Since I have no idea and can never know for sure, I don't cling to one answer. I like to imagine the possibilities.
Maybe it just ceases. Our brains turn on and we become conscious when we are babies, and they turn off and we cease to be conscious when we pass on. Maybe we just ceasing to exist, completely, as we ceased to exist before we were conceived.
I don't know that I think the consciousness is a figuratively tangible thing that must exist regardless of whether or not it has a vessel. But then, maybe I'm wrong! Maybe that's exactly how it is, and my "consciousness" will hang around in limbo waiting for another vessel to jump into. I think that would the be 'past lives' theory.
But then, who knows?? :)
This is a great conversation topic :)