It has been suggested that the reason that so many people relate consciousness to quantum mechanics is a sort of conservation of mysteries: consciousness is mysterious, quantum mechanics is mysterious, so maybe they are the same mystery. While the connection between them is admittedly circumstantial, they are mysterious in similar enough ways that we may speculate that at the very least quantum mechanics is a promising place to look for consciousness in the natural world. (See Seager (1995) for a similiar line of speculation).
First, we seek a place for consciousness at the very lowest levels of nature, and quantum mechanics is the lowest rung on the ladder, as low as our understanding of the natural world goes. It is the layer of inquiry of which we know only the behavior of the things we study, but we can not, in principle, know the intrinsic nature of whatever is doing the behaving. No one knows what an electron really is, beyond our ability to characterize its behavior as described by the relevant quantum laws.
Second, I have argued that at least as striking as the qualitative nature of consciousness (what is it like to see red?) is the all-at-onceness of our thoughts and perceptions, their intrinsic unity. We have first-person, immediate evidence of mid-level inherent individuals in the world (that is, inherent things that exist between the scale of the universe itself and, for example, an electron), and we need to square that with physics somehow. When I look at or think about a suspension bridge, my bridge percept is absolutely, inherently, a single thing. Physics is often taken as insisting that any such mid-level things are really heaps of low-level things. Just as heat just is mean molecular kinetic energy, I just am the pile of atoms that constitute me.
Quantum mechanics, however, gives us some counter-examples to this way of looking at the world, just as consciousness does. The very strange world of quantum mechanics is populated by bunches of things that come together to form one larger thing that can really no longer be thought of as a heap of separate components. In a quantum entangled system consisting of two particles, for example, we have multiple parts coming together to form a thing that is inherently, absolutely, one single unitary thing. Bose-Einstein condensates are another such example. In a Bose-Einstein condensate, the component atoms lose all individual identity, and the entire condensate is one single thing, with one single wave function. It is simply incorrect to think of a Bose-Einstein condensate as being composed of individual atoms anymore.
As with our percepts, a quantum entangled system is one thing, not an aggregate that may be seen as a thing when looked at or analyzed a certain way. The ontological reductionism inherent in a classical or Newtonian view of the natural world means that consciousness simply can not find a home in a world that is exhaustively described with such a view. Because quantum mechanics sidesteps this reductionism by providing a real basis for holism in the universe, by process of elimination, we ought to strongly suspect that consciousness and quantum phenomena are somehow related. See (Silberstein 2001) for a discussion along these lines.
Third, there is the problem of the alleged causal closure of the physical world, and the way quantum mechanics and the holism it implies allows us to wiggle out of it. The argument is often made that the laws of physics are airtight, that (assuming they are true) they account completely for everything that happens in the world, leaving no room for consciousness to have any measurable effect on anything. Unless, that is, you define consciousness strictly in terms of physical dynamics in the first place, which is to say that you subscribe to physicalism (and thus, in my opinion, define away the interesting questions and properties of consciousness).
It certainly seems that the laws of quantum mechanics are true, and dead-on accurate. The loophole in the causal closure argument may be that while accurate, the laws of quantum mechanics all yield probabilities only. They specify a distribution curve, not precise predictions - they predict collective behavior with 100% accuracy, but are agnostic about individual behavior.
If you run a quantum experiment 10,000 times, you are assured that your outcomes will fit this curve exactly, and for any one trial, the probability of one outcome over another is determined by the curve, but quantum mechanics is famously unable to tell the specific outcome of a particular single trial. It is an inherently indeterministic theory. Moreover, it is generally accepted that this indeterminacy is not a flaw in the theory or evidence of its incompleteness, but a fundamental feature of physical reality itself. No matter how well you know an electron's initial conditions, once it is in flight, you can not predict its position before you measure it. This is not because of any practical limitation on our ability to characterize the initial conditions of the electron, or any inaccuracy in the theory, but because the electron can not properly be said to have any definite position before you measure it. The position of the electron before you measure it is literally unknowable. It has only a likelihood of being in one place, and a different likelihood of being in another place. So the best theory we have about how the physical world behaves and most interpretations of that theory are, when it comes right down to it, indeterministic about the precise behavior of the physical world at a low level.
The only possible exception to this is the possibility that there are some kind of as-yet undiscovered "hidden variables" at work, and once discovered, they will allow us to predict the electron's position once more with Newtonian accuracy. Albert "God does not play dice" Einstein spent a great deal of his later life looking in vain for a hidden variable theory. Very few people seriously entertain the possibility of hidden variable theories today. Such theories are regarded as a philosophically (rather than scientifically) motivated attempt to restore determinism to the physical world. Even in the classical (i.e. non-quantum) world, it is becoming more apparent all the time that chaos and non-linear dynamics are the norm. Tiny differences at a low level get amplified to huge differences at a high level. There is no reason, therefore, that we would need particularly large-scale quantum phenomena to have a substantial effect on the macroscopic world around us.
It strikes me that there are many different actual outcomes of a given set of trials of an experiment that would still perfectly fit a given distribution curve, and thus not violate any laws that were given strictly in terms of conformance to such a distribution curve. The statistical distribution of letters I type on a keyboard might be the same whether I am typing a sonnet, a recipe, or meaningless gibberish. A complex coherent pattern may have the same statistical distribution as random noise - indeed, any maximally dense information, (i.e. maximally compressed information), is in fact statistically equivalent to random noise by definition. The door is open, at any rate, for patterns to result from the behavior of quantum systems whose coherence is not predicted by quantum theory, but which nevertheless does not violate the predictions that quantum theory does make.
In the well-known two-slit experiment, we don't know which slit the photon will go through. No simulation of the photon (on a computer, say) could tell which slit the photon will go through. The photon's behavior is not predictable through an analysis of its parts, because we must consider the photon to be one indivisible thing. The only possible way to know what the photon will do would simply be to be the photon. While the concept of free will could use some clarification, given the prerequisites of free will that I have argued for, I think it would be very hard to define free will in a way that granted us free will but denied it to the photon, given these facts.
Quantum mechanics allows for the existence of high-level entities that are causally efficacious, and whose behavior, while constrained by other entities, has an element that can only be called "random" by our best third-person physical theories. This is exactly the sort of physics that I have argued we must have in a universe that allowed for free will to exist. If quantum mechanics is a proper toe-hold for free will in the universe, then I am speculating about a hidden variable theory of sorts. Quantum indeterminacy is real, in that the behavior of quantum entities is unpredictable, and it is random, as far as its statistical distribution goes, but in another sense it is neither of those things: quantum entities behave the way they do because they must, according to their own laws. From a third-person perspective, outside such entities, their behavior must appear indeterminate and random.
I do not know if Stuart Hameroff is correct in his claims or if his research will bear fruit, but if I had to lay odds, I would speculate that something along those lines will turn out to be the case. It would not surprise me if at some point in the future it is discovered that the brain's activity depends crucially upon quantum phenomena, which are amplified to the level of neurons firing. Of course, the operative word here is speculate. It is worth noting that it is only under certain special types of circumstances that quantum systems can evolve in a state of entanglement or superposition without decohering or collapsing back to a classical state (leaving aside the philosophical thicket of the measurement problem). Under ordinary circumstances, we do not see quantum systems of any great scale (I avoid using the word "complexity" because it implies precisely the wrong thing, namely that a quantum system is made of parts, and that there may be fewer or more of those parts). So like Hameroff, I would suspect that we will eventually find structures in the brain that would support some reasonably large-scale quantum superposition which implies isolation from the surrounding environment. Whether the tubulin microtubules constitute such a structure, I am far from qualified to say.