I think it doesn't "shut down", maybe fades away to a different, low-power mode. When you go to sleep and then wake up, here is still a continuity, because of the same underlying structures and processes from which the consciousness emerges. So it is like a 4D thing.
That continuity never really breaks (until death? which is like singularity) and I think this is what makes you "you". You can't copy/teleport it (kinda by definition), but you can extend/transform it.
Perhaps a "Ship of Theseus"-style approach would work — gradually replacing neurons one by one, or brain sections piece by piece. Alternatively, the brain could be extended with peripheral "devices" connected to it with bandwidth comparable to the interconnections within the brain itself, up to the point until the biological part becomes irrelevant. This is similar to how a portion of the living brain can be removed while still preserving consciousness — as neurosurgeons often demonstrate.