Sorry, I added the example of the SF story while you were replying to my post. In that story now there's two of the same person walking about, both of whom think they are 'him'.
The really disturbing this is, this might be what our lives are actually like. After all, our brains frequently stop being conscious and then summon consciousness back into being. Is it really the same consciousness though? Or is it a fresh newly synthesised process that just has memories from the last time your were conscious?
I still wouldn't use the teleporter though.