The problem I have is that consciousness is one of those undefined words. Like does it mean being awake? Then those people who say "By the same logic, you die every time you sleep!" have a leg to stand on. But that's not what -I- mean when -I- say the word or read the word, I think in terms of having a personal experience, (and in terms of brain uploading) a logically-continuous personal experience. Dreams are still experiences, and even if I don't remember all my dreams, I still probably had them! And even if I had amnesia, I'd still likely feel like I am, at the core, me. The same person writing these words would be that same person, even if they don't remember ever writing these.
And this STILL doesn't get at what I think consciousness is. That all feels like it's beside the point, that it's talking about something irrelevant. More importantly to me, my idea of the underlying action of continuous consciousness, the thing that connects one moments "I" to the next, is the never-ending loop of activity in the brain. One neuron talks to another neuron talks to another, x1e10 in terms of complication and connections. Once you break THAT loop, THAT'S when I feel one person dies and another person begins. Whether it's destroy-and-recreate teleportation, or scan-and-recreate brain uploading, or brain death followed by resuscitation, there's still that break. Technically this isn't consciousness! I'll admit that! But it's what I feel makes a person a person and not another person who think's they're the original person.
That's why I support brain uploading in terms of connecting the computer to the brain and allowing the brain to migrate in over time, or replacing constituent parts of the brain over time either chunks at a time, or nano-machines converting neurons one-at-a-time. The loop of communication never breaks, it just starts talking to things that aren't neurons, and eventually that loop consists entirely within the machinery. It's still got loop-continuity back to the meaty origins.