I tend to envisage my mindscape as an orchestra. My consciousness is a fictitious conductor. It doesn't exist, but the lie that it does makes it easier to coordinate things between the instruments. In some manner, by acting on that lie, it is no longer a lie.

In this analogy, when the brain hemispheres are separated, then the orchestra is split in 2. Both develop a conductor, to try and remain functional. Neither conductor is the original me, but neither is not me, at the same time. It would be unpleasant for the variant left unable to communicate however.

I've actually experienced something that felt close to this before. A combination of sensory overload, and panic attack. My mind momentarily became completely discordant. As it sorted itself out, my consciousness reasserted itself in several different loci. In effect, my orchestra had 3 different conductors. It took almost a minute for them to stop pulling against each other and meld into 1 again. I have memories of all 3 sides in the 'battle'.