This is exactly the kind of stuff I think about all the time! I loved reading Moffat’s take on it.
What do you think: if you are teleported, and you are reconstructed somewhere else with different atoms, is it still you?
Someone mention Trigger’s Broom!
Personally, I’m of the opinion that your memories are what make you you, and as such a teleporter that creates a perfect physical copy and uploads your memories to it wouldn’t be killing you.
Semi-related, I was listening to the audiobook of Oathbringer from The Stormlight Archives, and there’s a conversation about a fantasy religion’s beliefs in there.
Basically the idea these people believe in is that when you go to sleep, your soul dies and another soul takes it’s place, inheriting memories and everything from the previous soul
You can’t exactly prove it untrue, but it’s also a belief that doesn’t really change anything in how you go about you life
But then there is the argument that if you make a perfect copy of someone and copy your memories into them, are they now you?
You’re both you and yet you’ll become individuals over time
The person I am isn’t the same as the person I was last year but they’re both me, if I had a perfect clone, we’d be the same person, but come next year we’re different people, even if we used to be the same
Yes, but probably with body dysmorphia. This is how I view body-swap stories, too.
This is actually precisely what I was reading about for my philosophy course yesterday, and then of course I tried to connect that to doctor who, I’m going to practice some arguments from that later here
I was about to comment on how philosophical this was all getting.
Oh yes it’s a very philosophical concept.
If teleporters ever exist then it would be easy, as long as you have enough atoms, to use them as a human photocopier.
Imagine the implications of that!
Just thinking about it, it definitely is not the same person coming out of the teleporter as went in.
It destroys your body and recreates it somewhere else. So you are effectively dead, and some new person goes “sauntering away” with a copy of your body and memories.
And how do you even transfer memories? Are they built into your brain, like in a computer?
This would mean that if you had teleporter technology you also have perfect mind reading tech.
Sorry I’m overthinking this maybe but it fascinates me.
The Thirteenth Doctor says she is thousands of years old, but the Fourteenth Doctor says he is Billions of years old. Does this perhaps reflect Chibnall and RTDs different approaches to this very dilemma vis-a-vis Heaven Sent? 
Hmm…
Well a teleport takes matter, converts it into energy, and that same energy is reassembled as the same matter. So no, it doesn’t destroy the original, just changes its form and turns it back again. The matter that is moved is in the exact same form it was when it left. Teleportation does not kill the traveling person, and it is the same person when they arrive.
Like the Bio-Printer in Red Dwarf.
I think that varies wildly from one sci-fi series to another. Some places it’s more like your body being broken down and utterly destroyed, your memories and “body print” are then sent in a sort of email to basically an organic 3D printer. Totally new body consisting of different matter but with the same memories…
Well, what is that bio print that is stored but again, energy? I think it’s still the same thing as I had said but with an extra step. I still argue same person and that they never were killed or destroyed, but again, changed form and back. And even from “product of one’s memories” logic, I still say same person, and they were never dead.
Seeing as how teleports aren’t an actual “thing” on this world we live in, but a product of fiction, it will of course be a thought experiment of sorts 
Fair. I’d also say, as long as there’s a carbon atom in every spot that had a carbon atom before, an oxygen atom in every spot that had a carbon atom before, et cetera with every atom you had before. Even if it isn’t the exact same carbon atom or oxygen atom, all atoms of the same element are the same. So I’d see it as the same matter. All of your brain and memory and everything is still just all the chemistry how all the atoms are lined up, so as long as the pieces are all put back how they were, same person.
Every cell in your body has been replaced many many times over your life. Does that mean you died and were recreated each of thoe times? I don’t think so.