Actually some theory on how consciousness works suggest that we as humans already have this. That we have several different streams of consciousness that work in cooperation with each other.
I read some texts on this some years ago. One of them described how when we try to solve a problem and get stuck, and then we go do something different for a while, and then suddenly we realise how to solve the problem seemingly out of the blue.
What is suggested to happen here is that one or more streams of consciousness recognise a problem as important when we struggle with it and it takes it upon itself to look for solution. Then when it finds a solution it presents it to the the rest and we feel like we've had an epiphany.
This makes sense to me because it makes the brain capable of working on complex problems while other parts are free to keep alert of the surroundings and ensure survival.
I'm not sure if the streams of consciousness that we experience as the self that is alert to its surroundings stay the same. But then again I tend to think about the sense of self as an illusion anyway.