Whilst I think the 'science' in this episode was pure nonsense magic, and evoking conservation of energy there was particularly painful, I don't think that working of the mind is really understood on the level you imply at all. Sure, we pretty well understand how brain works in mechanical sense, how certain stimuli create certain reactions. However, no one really has a fucking clue why we actually subjectively experience any of this. As far as physics go, we all should be just robots, philosophical zombies, yet we are not, we have qualia, phenomenal experiences. No one has even remotely satisfactory answer for this Hard Problem of Consciousness, as Chalmers called it. I'm of course not implying existence of souls here, but merely pointing out that this is a subject which directly involves one of the biggest, if not the biggest, unanswered questions we have.
Even given all of that - the possibility that the "thisness" of experience might be something which can't be quantified yet - it's well established that most of the things that we bind up closely in our sense of self - are part and parcel to the brain. Memories are stored there, for example, and our personality being what it is, instead of something different, is because of the particular physical structure of the brain that we have.
Thus even if I did have some sort of ineffable "experiential point" which could be sucked out of me and put somewhere else, if this continuity of consciousness didn't come along with my memories and my personality, no one else would define it as being "me." They'd say the p-zombie I left behind was me instead.