If even the new consciousness isn't able to know, then what does "same" mean and what part of that definition aren't you satisfying? More acutely, why should you care about satisfying it?
Perhaps a good alternate angle is to consider the “copy” (rather than “replacement”) proposition someone else mentioned: let's say a perfect copy of you could be made (i.e. the entirety of a consciousness is somehow material). Are the two consciousnesses linked somehow? Or are they completely separate despite thinking exactly alike, having the same memories &c.? If they are separate, what happens if the original is killed? Then, the next logical step is to ask how that is different from a consciousness transplant.
It's wild stuff. Or seems like it to a thus-far (presumed) single consciousness.
Selfless, but unintentional.
It'd be another thing entirely if you could join the 'threads' back together, though. Imagine forking yourself into 12 different entities, each living a separate life for 100 years, then reintegrating.