Some people are reported to have survived with only one hemisphere of the brain. It's not a huge stretch to imagine that you could transplant into yourself all needed duplicates or artificial replacements for vital organs, then split yourself in two so both halves survive, with one of the brain hemispheres each.
Then, there are two new independent yous there, neither of them being quite like the old you, neither of them feeling the other you as yourself anymore, but both of them independently feeling themselves as you. Stop and imagine this situation. How do you think you'd feel? How do you feel now, thinking about it?
Now let's leap a bit from the medical to the philosophical: let's abstract the fact that parts of the brain are specialized. Assume that we could do other partitions but left-right, or suppose that brains will rewire to make up for lost functionality. Imagine it's actually true that 90% of our mental power usually goes to waste, and that we could lose half of the brain and somehow mostly make up for it. Or imagine that it's possible to make a perfect copy of yourself, as proposed elsewhere in this thread. The result is the same: there is another you out there, virtually identical, with the same past, but which you don't feel as you anymore.
How do you feel about that other you?