Hilary Putnam once held the position (i.e., in the 1960s) that consciousness is literally the brain computing. That also had the implication (for some people) that it's not necessarily the brain that causes consciousness: it's computation itself. This is the case even if it's computation of a particular kind (in conjunction with its physical or biological implementation) that's required for consciousness. Thus it followed from this, according to Putnam, that if computation is everything, then consciousness can occur in a computer or even in Putnam's own “brain in a vat”. Again, what matters are the computations, not the physical basis of those computations. (To be accurate, in his 1967 paper, Putnam hardly refers to consciousness as such. He talks about “mental states”.)
Daniel Dennett also argued that the computations of the brain are consciousness. This doesn't mean that such brain computations result in consciousness in some kind of causal and temporal sequence. It means that when the brain's computations occur, so too do conscious events. They're one and the same thing. Or, as Dennett himself puts it, it's not a case of brain computations occurring (in what he calls the “Cartesian Theatre”) and then we become conscious of those computations. Instead, when brain-computations occur they actually constitute consciousness. When the brain computes: that's what consciousness is. That's what we're aware of when we're conscious; though we aren't necessarily aware of them as computations (or as anything).
This is similar to John Searle's view in which he says that there's no causal and temporal sequence from a brain state to a mental state in that the mental state is the brain state. The brain state and mental state occur at one and the same time. It isn't a case of brain state at time t1 causing mental state at time t2. Both the brain state and the mental state occur at t1.
The mental state is the state as experienced from the inside (as it were); whereas the brain state could (in theory at least) be observed by neuroscientists. However, it need hardly be said that Searle doesn't believe that this has anything to do with computations (as such). In his case, it's a point about the (causal) relations between the brain and consciousness; not between computations and consciousness.
Finally, it needn't be the case that consciousness is always (or at all) computation. We can substitute all the references to 'computation' in the above with the words 'brain event' or 'brain state'; as Putnam himself did in his paper of 1967.
Dennett, Daniel. (1991) 'The Cartesian Theater and “Filling In” the Stream of Consciousness'.Putnam, Hilary. (1967) 'The nature of mental states', in Mind Language and Reality: Philosophical Papers, Volume 2.
Searle, John. (1997) The Mystery of Consciousness.