Transporters work by de-assembling something (e.g. you) and re-assembling it somewhere else. What if, when you’re dis-assembled, you die, and the re-assembled version of you is essentially a copy? Then every time someone steps onto a transporter, their final thought before death is that they’ll end up beamed somewhere else. And the re-assembled version (copy) just thinks that everything went fine and continues on like nothing bad happened.
Haven’t watched Star Trek in any form, but I think I’ve heard of this thought experiment before anyway.
The way I see it: “the you” at the transmitter will die. There’s no place for you to go as your body is torn into atoms, unless you start bringing in “souls” or something. The question in the more philosophical sense is that, suppose you can get a 100% identical clone down to (almost) continuity of consciousness… is that still you? Really you, whatever that means?
Speaking of which… what if you left out the “breaking down” part? Are there just two of you now?
That last part about there being two of you reminds me of The Prestige. I won’t go into further detail in case I’d be spoiling it for you, but would encourage you to watch it if you haven’t.