But today's you has diverged from yesterday's you just as much as tomorrow's copy would have diverged from today's original. At the moment of copying - assuming a sufficiently high fidelity - they would be practically identical. The differentiation would only happen over time.

So if you kill the original - maybe because the procedure is destructive or because it's a consciousness-saving emergency measure - the "copy" would have a pretty good claim to be a continuation of you.

But why is "your experience of the world" tied to your current physical body? I think that the brain is just a machine that creates the illusion of a consciousness. If you turn the machine off, copy the all the data into new hardware, and turn it back on, it will start back where it left off, and the "you" inside won't know the difference. You might not even notice it happened

Agreed. Responding to the replies, regarding the fidelity of a copy: Yes, the copy would have a great claim to being you. Assuming the physiology of the copy is also (nearly) identical (behavior and personality is not a pure function of the brain), the world at large would also identify it as you.

But I'm not interested in legacy style immortality. I don't care at all if a separate version of me continues to exist – I want to be continuous with the physical substrate I now occupy. In other words, we are all Ship's of Theseus, and that's fine, but a perfect replica of the Ship breaks my internal narrative of continuity.

To distinguish: If a form of "immortality" supports the possibility of another copy of me existing simultaneously with the current incarnation, then it is not the immortality I'd desire.

Responding to the parent, I would much prefer slowly replacing my neural hardware with circuitry to wholesale scanning and instantiation in another substrate.

Summarize: Identity is (partially) a function of substrate. Change the substrate in a discontinuous fashion, and identity itself becomes discontinuous. I call that discontinuity (one form of) "Death".