Is Consciousness Continuous?

Artificial intelligence brings up interesting questions about our own. If we assume an AI has consciousness, it's natural to ask if it still has the same consciousness after being turned off and on again, and if so, if it's still the same after being transferred to a new computer. Yet for some reason we've never asked ourselves this question about ourselves because to us it's always seemed as if we're the same person after waking up from sleep, coma or near-death.
Since consciousness in either case is made up of electricity in the circuits, it seems like once the electrons are gone, so is consciousness. Granted, the electricity in our own circuitry never fully goes away until we've started to decompose, but then again, it's made up of a large number of patterns that aren't always connected, so which of these is our consciousness in, and what happens when that pattern stops while another begins? How about someone with dissociative identity disorder, where the patterns have dissociated into entirely different personalities? When an alter is taken over by another, does their consciousness stay the same, and if not, does it return when it takes over again?
We take a lot for granted in life. Perhaps we shouldn't even take for granted being alive. Perhaps we're only here for a few moments. There's no way to know because we can't remember having been the consciousness that came before: the only thing we know is that we are conscious right now, and we don't know for how much longer. Maybe the next moment my consciousness will transfer into someone else's brain. Considering this thought experiment can be a good exercise to live in the moment.