Now I could pretend that I am using this topic as a springboard to open a discussion about the nature of existence, consciousness, what makes you "you" and all that stuff.
But who am I kidding? I'm just nerding out.
You are a starship captain in the Star Trek universe. You take your first nervous step onto the teleporter pad, and wait for the engineer to throw the lever, instantly transporting you onto your new starship. What happens next?
To an outside observer, you are surrounded by a wispy cloud of light, you disappear, then reappear in a wispy cloud of light at your destination. Simple.
But what about your own personal experience of the voyage?
Believe it or not, teleportation is real. Admittedly we can only teleport single photons of light, but we can do it. But the process involves destroying the original photon and recreating it at the other end.
Let's imagine that this is how Star Trek teleporters work. The teleporter machinery reads you down to the last individual quantum particle, destroys you, then recreates an exact replica at the other end, complete with all your memories leading up to the moment of teleportation.
The you at the other end has only just been created literally a second ago, but it has all of your memories, so according to this new you, it experienced stepping into the teleporter, being surrounded by light, a brief moment of nothing then back to being surrounded by light at the other end. But the old you experiences stepping into the teleporter, being surrounded by light, then all of a sudden finding themselves standing at the pearly gates saying "I don't remember entering these coordinates!"
It's an easy enough mistake to make. The very first human being to be transported pops out at the other end and says "yep - she works a treat!" not knowing they had just been killed. So then everyone thinks it's safe and zips here there and everywhere by transporter, creating clone after clone of themselves that simply remembers a nice quick transporter journey.
So every time Captain Kirk is beamed down to the surface, the last thing he remembers is stepping into the transporter beam. Then when the new Captain Kirk beams back to the ship, the last thing he remembers is being engulfed in the transporter beam. And so on and so forth, with an endless parade of Kirks whose lives all end on the dispatch side of a transporter journey, until the very last time he ever transports anywhere, and his last clone goes on to live the remainder of the original Kirk's life with countless memories of being safely transported in the transporter here there and everywhere until he finally dies of natural causes instead of instant transporter beam disintegration death like so many Kirks before him.
Even if you could find a way to transport the original matter, so that no-one's body was being destroyed and then a copy created, it would still involve ripping you apart atom by atom, flinging you across space then putting you back together again. A process that would undoubtedly kill you anyway. In which case the first person to be transported would reach their destination and instantly flop on the ground like a marionette with it's strings cut. Which would then lead them to the quantum teleportation idea, which everyone would think is safe, and refer the previous couple of paragraphs for why this would be a problem.
It's the problem of continuity, and it's the same problem with uploading your consciousness to a computer. The uploaded version of you is still just a copy. The original you is still in your body. You could probably even have a nice conversation with yourself before you keeled over from whatever disease you had that you were uploading your consciousness to try and avoid.
It's a problem that was handled nicely in the movie The Sixth Day, when Mr Bad Guy clones himself and the clone wakes up before the old him is dead.
There are some people who don't see a problem with this. "Hey" they say, "if I still exist in some form or another, so what if I'm not around to experience it?" But, um, I kind of have a problem with that, because, you know, I don't want to die in a transporter room, or spend my last few seconds watching how much fun my digital copy is having without me. Besides, we can already do this kind of thing anyway - it's called having kids.
So now we could go into the whole thing of "but what makes you you?" and "where does this idea of me come from?" But...... let's not.
For now I can confirm there is no way I'm stepping onto any transporter machine or uploading my consciousness to any digital heaven until they sort these kinds of problems out.
And to answer your question, yes, this is the kind of crap I actually spend time thinking about.