Some philosophers think that every time you lose your train of thought (e.g. zoning out while driving or falling asleep) your consciousness disappears and a new consciosness forms later when you "snap back" (or wake up). This new consciousness shares only memories and personality with the previous one, but it is otherwise a new entity, similar to the teleporter thought experiment.
Conveniently, this makes death less scary.
Further reading: Zen and the Art of Consciousness by Susan Blackmore
I think what I was getting at is that it may be impossible to tell from the outside if something is conscious or not. If I can't even prove to myself that another person is conscious, I might have a hard time proving that a computer is conscious.
It might depend on how you define "person". When I look around, I see distinct human bodies. When I wake up, I'm in the same body as when I went to sleep, so in that sense I'm the same person.
What if "I" woke up in someone else's body? I'm sure there are movies with this premise, where "someone" wakes up in someone else's body, but they have the memories from their previous body.
But if memories are just physically encoded in the brain, I don't think this scenario makes sense even as a thought experiment.
When "I" wake up in the other person's body, I would have all their memories and none of my previous memories. So I wouldn't even know.