If OP was a seinen, I think it would be interesting if Mihawk was a bastard child of Imu. Alot of Imu's actions in the latest chapters, like referring to himself in the third person, cutting up the posters of the Ds, and ranting about the poneglyphs makes me think that Imu can be irresponsible and somewhat childish, and the first character that came to mind when thinking about rulers like that would be Robert Baratheon lol. That would explain the eyes.
Another theory that I have is that Imu's family, the Nerona family came from the same island as Mihawk did. Mihawk gives me former royalty vibes. The Nerona family could have intermarried with the Dracula family before abandoning the kingdom for Mary Geoise, which would make Mihawk an indirect descendent of Imu.
I'm not sure if the Vivre Card revealed what sea Mihawk was born in, but he lives on an island that's a ruined, abandoned kingdom. We know he was betrayed at some point too.
Your speculation about awakening being two souls in one body is interesting. Maybe only mythical zoans have red eyes when they awaken?