The consciousness of coma patients doesn't continue to exist because coma patients are, basically, unconscious. You may be thinking of individuals in a vegetative state, who exhibit varying degrees of it, but for various well-defined reasons cannot translate inner experience into outer expression.

There is no such thing as a briefly brain dead patient, either. There is a specific definition of brain death that has to be observed using specific clinical symptoms, and brain death is permanent by virtue of the signs used in its diagnosis. You may be thinking of people who meet the clinical definition of death temporarily and who are revived. In that case, their brains are often doing some very interesting things with their lack of oxygen. (I have experienced a smaller version of this; I am prone to hypotensive states if I overexert myself or get blood drawn and I experience some really bizarre symptoms because blood is not getting to the parts of my brain that it needs to get to.)

I think the issue with dealing with our anecdotal, experiential understanding of consciousness - c.f. why do we see the low end of the electromagnetic spectrum that we can see as red - is that we put too much emphasis on 'oh, we see it as red and not purple' and neglect the fact that this is how our brain is making sense of the difference between two wavelengths (and how about people who have different kinds of colorblindness? Surely the qualia people have considered that there are people who can't distinguish red and green).

I mean, fundamentally our brains are electrical meat and there should be nothing mystical about it.

EDIT: RationalWiki on qualia - http://rationalwiki.org/wiki/Qualia