We cannot compare the complexity of objects; we can only compare the complexity of our models for these objects.

And when we talk about models, the complexity of a model increases when our understanding of the object increases. Given enough resources and motivation, we can research an object forever.

Observations and research of the human brain predates the written history. So does the knowledge of the human nature and behavior. Everyone needs and learns at least some of this knowledge.

On the other hand, the research of the Sun internals is rather recent - as much as few centuries and limited to a small group of physicists. We have quite a clear understanding that we are not in a position to control anything that happens in the Sun, so our motivation is somewhat limited to almost pure scientific interest.

To sum up, our models of human brain are much more developed and much more complex than anything we know about the Sun.

This answer started as a comment to @Wolphramjonny who remarked that life emerged from evolution which the sun does not undergo.

My first objection is that the universe as a whole, and thus the stars in it, could conceivably be subject to some kind of cosmological evolution, as proposed by Lee Smolin.

My second objection is that the Sun and its internal structures do provide some of the elements needed for something that could be called life in the broader sense. If we try to assess alien environments it's probably prudent to keep an open mind. We must consider the possibility of life that is not carbon based and perhaps doesn't even need a familiar material substrate.

We can, instead, try to specify a few abstract notions of life. Any conceivable life, independent of its substrate, will be

1. a complex system
2. whose sub-structures pass information spatially, i.e., communicate, thus forming the interconnected system in the first place;
3. that can preserve and pass information temporally, thus establishing identity;
4. that profits from some energy flow which allows it to establish islands of low entropy to store its curated information;
5. that is subject to some kind of evolution.

In a cell and in multi-cellular organisms we can easily identify these traits; ironically in this context, the energy flow we profit from is exclusively solar (even geothermal heat and nuclear fission utilize solar energy from past supernovae). Now let's assess the Sun.

1. In the sun we can see some degree of complexity, but it probably doesn't reach the complexity of the brain. If we restrict ourselves to sub-structures like the granules we have a few million of them: The typical convection cell has a diameter of 1000km. That gives it a surface area of $10^6 km^2$. The Sun has a surface area of $6.07 * 10^{12} km^2$, accommodating roughly $6*10^6$ convection cells, which is the numerical complexity of a simple cell. Whether there are any meaningful smaller structures is unclear.

2. I was at first skeptical about the communication between different locations of the sun but after revisiting some videos like this or this I must admit that there is actually a lot of interaction between different locations; think of the plasma arcs and magnetic field "bundles" as axons.

3. The hardest problem will be how to preserve information in an environment as hot and dynamic as the sun. What we would be looking for are cooler islands in the bubbling soup, ideally longer-lasting ones. Like the permanent coronal holes close to the poles. Interestingly, these holes come and go in 11-year generations. Honi soit qui mal y pense. ;-)

4. There is certainly a lot of energy flow available in the sun; there obviously is a considerable temperature gradient and corresponding heat transfer from the core to the surface.

5. Evolution is often defined as the combination of procreation, mutation and selection. In this sense some natural phenomena that are not alive can be seen to "evolve", especially dynamic systems like dynamic fluids with waves and vortexes, or other (e.g. electromagnetic) waves, or Conway's Game of Life.

   The Sun falls squarely into the category of (highly) dynamic systems with a large number of discernible internal structures, like convection cells. The structures in its interior procreate, change, and hell yeah, there is a selection mechanism. But is the selection mechanism too harsh? Even with an open mind it is hard to see how bubbling plasma even in coronal holes can provide a nourishing environment for structures stable enough to obtain some kind of identity — not in the sense of self-awareness but simply in the sense of a discernible information storage that exists long enough to evolve at all. As others have said, the convection cells are relatively short-lived and probably fall short of this requirement. Coronal holes last much longer. But we would still need some kind of information handover from cells or holes to their successors. Whether smaller, yet unknown structures could be stable enough to start "living" is unclear; the environment in the sun is clearly very destructive.

Summing up: The Sun has a number of known structures reaching the numerical complexity of a simple cell. The structures do interact, and some of them have life times of a year or so. The key question is whether it is possible to preserve information and communicate it under the harsh conditions of bubbling plasma. Even with the somewhat tongue-in-cheek argument about coronal holes it seems highly unlikely, barring any unexpected discoveries. It seems much easier to harness this energy flow farther away, in the habitable region, like on Earth. Of course unexpected discoveries are possible, even likely: We don't even know our own planet very well, let alone the Sun.

Memory

When we think about what it means to be "conscious", I hope we agree that memory is an essential element. "Information processing" is also an essential element, but many artifacts possess this capability without any long-term memory. For instance, the read/write head of a hard disk may be managed by a PID controller, which clearly has sophisticated information processing capabilities. But I doubt that anyone besides Rupert Sheldrake and his acolytes would say that it is "conscious". Rather, most folks would say that the controller performs its job "mindlessly", not least of which because it requires no long-term memory of its performance, no creativity, no improvisation. Its behaviour, while dynamic at the micro-scale, is fixed at the macro-scale.

Similarly, if you met a robot that could speak conversationally in your first language, but could not remember anything more than 2 minutes old, then you would probably be skeptical of any claims that said robot/AI is "conscious". It may be claimed that it is equivalent to a human who has lost the ability to form long-term memories, but I would argue that such people, tragically, have lost some measure of their "consciousness".

Of course, the brain stores memories in its connections: the synapses between neurons. However, it doesn't store them like bits of data in a computer's Random Access Memory (RAM). The information is stored both by the strength (weight) of the connection, as well as by the topology of the network itself. The where of the connection is just as critical (if not more so) than the how much. And this implies that for the brain to store memories over the long term, it must have a stable topological structure. This is, naturally, why CNS neurons do not, as a general rule, get replaced over your lifetime. New neurons may grow, but they do not "take over" the role of neurons which have died.

The brain is able to maintain its topological structure, because it exists, for the most part, in the solid state. Neurons are slippery, squishy devices, with a significant liquid state component, and the brain itself has the consistency of jello, being able to undergo significant deformations without loss of structure. Even so, if the brain were a fully liquid soup, where any portion could migrate to any location within the volume, then I think we would all be very hard-pressed to explain how it retains memory or function over any meaningful time scale.

The Sun

The problem with Rupert's argument is that he simply hand-waves away any details by appealing to "structure" and our ignorance, and pretending that putting the two together will lead to consciousness. But the big problem you have with the sun is that it is not solid. Nor is it semi-solid, like the brain. Nor is it a liquid, like the ocean. No, the sun, from its surface down to its core, is one blazing-hot plasma. And anyone who has tried to build a plasma-based fusion reactor knows that a plasma that is hot enough to fuse H, D, or T is so unstable that there is nothing like a long-term structure capable of holding a memory.

When we look at the environmental operating range of the brain, we see that it is a very fragile instrument. If it goes just 10 C above nominal, it can seize up and stop functioning entirely. While silicon processors have a much broader operating range, they are also much more robust by being literal crystals of metal. The intra-molecular bonds of silicon chips help them retain their function over a much broader temperature range. The sun has an internal operating temperature of about 15 million K. I think it is quite safe to guess that the spatial relationship of any two atoms within the sun does not survive long time scales (for just about any definition of "long time scale").

Now, this is again hand-waved away by claiming that consciousness is not, in fact, stored in the structure of the brain, but rather in the electromagnetic fields. This is just silliness. If having an "electromagnetic field" is sufficient to have consciousness, then everything (made of normal matter) has consciousness, including my coffee cup. And at this point, "consciousness" ceases to be a concept, because it is equivalent to "vwerp". You may never have heard of vwerp before (in fact, I really hope you have not) because I just made it up. It doesn't matter how you detect vwerp or whether it's interesting or harmful or anything else, because the one thing you need to know about vwerp is that every object has vwerp. And once you know that, you know it is an utterly useless and meaningless concept. Well, you also need to know that there is no way to distinguish the vwerp of one thing from that of another, because the vwerp is ultimately shared amongst all things. So it's not like position or momentum or other physical properties that we presume everything also has. It's universal and indistinguishable.

This clumsy sleight of hand is necessary to remove the topological structure of the brain as a necessary requirement for other conscious artifacts. It is the most obvious thing which makes brains different from coffee cups and clouds and ironing boards. But all of those have electromagnetic fields of some sort, so they also all have consciousness, according to Rupert. And that leads us to the Final Silliness.

Morphic Resonance

Rupert is not really interested in physics, so asking about the physics of the sun is really missing the point. He believes in a magic phenomenon called "morphic resonance". You can test this phenomenon by recruiting a friend, and having them stare at you when you are not looking. You then record when you sense being stared at, and when your guess is correct, you have confirmed the theory of morphic resonance! This theory has, in fact, been validated thousands of times...on the internet...by volunteers. It's science, right? If you think so, Michael Shermer would like to have a word with you. You see, some people...whom you might call "real scientists"...have also attempted to replicate the morphic resonance experiment, and failed. But, not to fear! Those scientists suppressed the morphic field with their doubt. So, in fact, it is impossible to falsify morphic resonance, because any attempt to do so will destroy the very phenomenon being studied!

So you see, it doesn't matter if the sun has large-scale structure or not. It doesn't matter whether plasma granules behave like silicon granules or whether the magnetic fields in sunspots are strong enough to store memories. Because at the end of the day, the morphic field is not about science. It's like asking whether a salty lake is an adequate test environment for detecting witches.

It has recently been shown that the Navier-Stokes equation (the equation governing the flow of a continuous fluid) has global solutions computationally equivalent to a finite number of steps run on a Universal Turing Machine, and thus can implement any finite computational process that a general-purpose computer can. Presuming intelligence to be Turing-computatable (still a controversial position in the philosophy of mind, of course), this would mean it is possible in principle for a fluid to have intelligence. Whether there is any feasible route for it to evolve from convective randomness is another question, of course, but the same might be said of a demonstration that the chemical reactions going on in a bucket of "primordial soup" are Turing-complete. If we "can't imagine" how a rockpool filled with random chemicals could evolve into intelligent life (or even store information over a few decades - not even as long as Jupiter's red spot has lasted), is that evidence of its impossibility, or merely a shortcoming of our imagination?

The plasma of the sun is of course not an ordinary fluid, but flows subject to magnetohydrodynamics, where the plain Navier-Stokes is coupled with electromagnetic fields and currents. But I rather suspect that makes it more complicated and capable of computation, not less.

We have a decent model of how the sun works. What I mean by decent is that it is accurate enough to predict 5 billion years into the future that the sun will become a red giant, and then ultimately a white dwarf. The model is based on well-understood concepts of gravity, strong nuclear force, buoyancy, and electromagnetic radiation, and its explanation satisfies most people. Even though people are still researching the sun, I don't think that anyone anticipates that there are huge missing pieces to the model that will radically change our understanding of how the sun works.

In contrast, I don't think anyone can satisfactorily explain consciousness. Psychologists have empirical models to predict behavior, but their predictions are usually just one move in advance, and are not 100% accurate.

Beyond functionality, I just can't imagine that the important aspects of the structural complexity of a star are anywhere near that of a cell, let alone an entire brain. The DNA in one human cell has information encoded in about 6 billion letters (3 billion from each parent) in an ordered sequence, encoding about 25,000 genes. There are 4 possible letters, so we are looking at one sequence out of $$4^{3\cdot 10^{9}}$$ One neuron makes about 1000 synapses with other neurons. There are about 100 billion neurons in the brain, and we know that cognitive function changes when the brain loses neurons.

The communication at the synapse between neurons gets weighted through learning models of neural networks. These and other levels of complexity in cells, such as how proteins fold to a minimum energy conformation, determine easily-observed aspects of the brain's behavior. I'm unaware of any comparable amount of complexity in a star.

Niels Nielsen's answer is excellent when it comes to applying the scientific method to the question. I'd like to add more to the discussion, when it comes to how we might try to quantify these things.

"Complexity" is a difficult think to define or measure. As you point out, the core of the question concerns the ability for celestial bodies to be conscious, rather than complexity, which is the direction I'll take my perspective. That being said, there is perhaps some relationship is between consciousness and complexity, as we'll get into.

Before we continue, we should point out the elephant in the room; which is the assumption that the brain is what generates consciousness. This is an open problem in the field of psychology. Regardless, a naïve way of measuring complexity could be with looking at if the entropy of the sun is higher than that of a brain? It is no doubt here that the sun has vastly higher entropy, but I can name a lot of other things that have higher entropy, but aren't inherently conscious. However, what exactly is consciousness?

American physicist Michio Kaku has a very elegant philosophy on the mathematics of consciousness (as explained in this video), in which consciousness is perhaps just a measure of how 'meta', or self-referential a system is. In the video, he gives the example of how a heater in a house has the minimal unit (or degree) of consciousness, 1 feedback loop. It affects the temperature which then affects it. The average human, on the contrarily, has literally billions of electrical feedback loops within its nervous system, not to mention other external feedback loops, such as the network of society. Thus, the measure of consciousness of a brain is at least on the order of billions higher than your household heater.

Continuing with this definition, at the very least, the sun has at least a few degrees of consciousness, concerning the feedback loops inherent with nuclear fusion, and the balance between gravity and runaway-heat. But are there any 'circuit'-like loops like that of the brain? Absolutely! Consider how the ions in the sun love to travel along magnetic field lines, giving rise to the phenomenon of the coronal mass ejections, and sunspots. Are these electromagnetic fluctuations more complicated the brains, however?

As a simple back-of-the-envelope calculation to get a rough idea of the order of magnitude, we'll count the number of smallest meaningful units when it comes to the complexity of the sun's surface. These are granules, and are an average of $1.5 \text{km}$ in diameter, which, assuming a circular granule, covers an area of $1.8 \text{km}^2$. Now, the sun has an enormous surface area of about $6.1\cdot10^{12}\text{km}^2$, hence, the number of granules comes out to

$$\frac{6.1\cdot10^{12}\text{km}^2}{1.8 \text{km}^2}\approx 2.6\cdot 10^{12}$$

Therefore, considering this calculation only took into account the sun's surface, it safe to say that electromagnetic complexity sun is at least a thousand times more complex than the human brain. This is roughly the same scale of magnitude difference from the human brain to the ant brain.

There is definitely many things overlooked with this analysis, and there are many problematic assumptions. However, I do think there is merit to large systems having some degree of consciousness, analagous to some notions of a Boltzmann brain. It is entirely possible that the sun may have the intelligence of an insect, or that of a demigod, but due to perhaps a lack of sensory input of its environment (except, gravitational influence), has no clue what's going on outside of itself, not to mention and entirely different (and to us, probably abstract) experience of existing.

I would like to mention something the other answers do not address, and that is predictability and higher level internal structure.

Fortunately, from our perspective, the Sun is (relatively, compared to the human brain) predictable. Its internal structure can be modeled with a high enough precision (thanks to quantum mechanics (QM)) that predicts its effects on our lives here on Earth.

This is not the case with the human brain, unfortunately. Our capabilities predicting the human behavior and the human brain's working is very limited (relatively to the case of the Sun).

But why is it so? Both the Sun and the human brain consist of elementary particles at the lowest level, and we do have quantum mechanics, so we should be able to predict how both of these work with the same confidence (like we can predict the Sun's effects on us here on Earth). In reality, the human brain has something, a higher level internal structure, consisting of neurons. Our capabilities, and regarding how the neurons work, are very limited (relatively to how we know how to describe mathematically how the Sun's internal structure works based on QM), especially their interconnectedness.

How Many Atoms Exist in the Universe?

Just to clarify, the number of atoms in the Sun is orders of magnitude greater than the number of neurons in the human brain. The problem is, we do not have a mathematical language that would describe the neuron's working (and we do not even understand it at that level) and interconnectedness like we have QM to describe the atoms in the Sun.

Is there any evidence in support of the specific claim about the complexity of structures within our sun?

Complexity of a physical structure is unfortunately a vague concept; it depends on our model of the structure, and the resolution scale we use to sample it.

The sun is very complex when we try to describe it with resolution of 1 meter - there are many things going on regarding convection of gas, magnetic fields, particle streams and so on.

The sun is even more complex when we describe it with resolution 1e-10 m. Immense number of particles interacting via EM and nuclear interactions.

The sun is pretty simple when we describe it with resolution 1 sun diameter. Hot radiating point of space with quite stable and isotropic output.

Is it even meaningful, within the language of physics, to talk about Structure A being more complex than Structure B?

It can be, if "structure" is a model of physical structure. For example, model of Sun with 3 parameters that are related via simple equations is much less complex than model of Sun that describes it as set of immense number of mutually interacting particles.

We can also compare complexity of models of different objects, when these models are of the same kind. For example, complexity of ant brain model that describes it as network of neurons is probably lower than complexity of the same kind of model of human brain, simply because there is less neurons.

But it is hard to do this for different models. We can't identify and count neurons on the Sun, and can't find convection cells and plasma acceleration events in the brain.

Of course, for two fully specified models in form of books we could probably estimate whether one is more complex than the other, by counting the number of pages. But would that be a useful thing to do? Any of the two models could be further enhanced and made more complex to win this complexity contest.