The outlined perspective is a composite consciousness model. Right now you have one big consciousness that is composed of many parts of your brain. If you split the brain then that splits one consciousness into two. This idea can help you sleep at night.

An alternative is that it is already like that... right now... you speak and the obvious logical conclusion from your perspective is that you said it. It could be that another subject, with experience, said it. Subjective you might be the nonverbal subject who has gotten so good at watching verbal subject that the words are so familiar, you know what its going to say, it may as well be subject you.
If verbal subject you says something stupid you nonverbally give it some feedback through the hemisphere connection. Once it's cut things go on, subject you just can't tell verbal subject when its an idiot anymore.

Uh oh.

Why stop there? Something akin to the Freudian Id, the Chimp Brain, the Lizard Brain, whatever you call it might also be in there with a subjective experience... serious mental disorders might actually be a struggle between subjects. A well integrated person might be someone where all subjects have learned to be on the same page and work together.

Uh oh. The word 'subject' and the word 'experience' provide you no relevant, additional information. Maybe, you are the captain of your own brain ship is a much better metaphor than you thought. Do you listen to feedback from your crew and do you teach them how to be more effective?

After all, that's the whole point of consciousness.