There are reasonable arguments for the incomprehensibility of human consciousness, and some of them were given here the other day in Comments. Let me offer arguments for the contrary.
First, one very important feature of consciousness has already been comprehended. We can say with a high degree of confidence that there is no ghost in the machine, that consciousness is an emergent physio-chemical property of the material brain. Whether consciousness is deterministic or involves some measure of quantum uncertainty remains to be seen, but I find Roger Penrose's argument for quantum uncertainty unconvincing. For the moment, Ockham's Razor rules.
Second, we can study emergent consciousness by observing other organisms, from sea snails to chimpanzees. That is, in principle, we can build up an understanding of human consciousness incrementally. This assumes, of course, that human consciousness differs from that of other organisms only in complexity, not kind. Again, for the moment, the Razor rules.
Third, as I mentioned here once before, a project is underway to fully map the neuronal structure of the human brain, at which point it should be possible to construct an operational electronic analog of the brain. Will such machines be conscious? Google "artificial consciousness" and you'll find arguments for both sides. At the very least we will pare away some of the incomprehensibility.
Fourth, we may already have created a "conscious" machine: the internet, which approaches the human brain in its degree of interconnected complexity. It is continuously "aware," sensitive to millions of sensory inputs -- touch, vision, hearing, smell, and for all I know even taste. I can ask a question in human language or tap an icon and instantly have a response from the internet's vast memory. The internet and its myriad of input/output devices mimic enough of the aspects of human consciousness for us to be increasingly confident that consciousness is not intrinsically beyond in principle understanding.
And isn't in principle understanding all we ask of science?