I have a world with days lasting 9 years. The inhabitants of this world have taken to migrating across the globe to avoid the deadly night, keeping themselves in a sweet spot. One of the biological quirks of the people living here is that sleep isn't a thing. Similar to dolphins, they sleep with only one hemisphere at a time, allowing them to maintain consciousness at all times. This allows them to get their required sleep while still being able to migrate throughout the 'day'.
As I began to develop the mythology and cultures of this world, a question popped into my head that would heavily affect everything this species does; how does their consciousness work. There are various signs that our right brain is really its own intelligent consciousness that works with our speaking left brain; But on a species where each half is only conscious for half the time, would they only be able to speak half the time? What about the other things?
The right half of your brain dominantly controls things such as spatial abilities, face recognition, visual imagery and music. While the left half deals with language, math and logic. Would a species that only uses half the brain at any moment only possess one half of these features at any given moment? Would it have two separate personalities? Or is there other things that I (admittedly not a neuroscientist) just am not seeing?