A smallish but somewhat complicated idea: The ability to actually talk to yourself, rather than it just being a silly message that pops up when you shift+c with nobody around.
I’m ok with it only working if you have schizophrenia, or psychotic, or such. Basically; mechanically have an NPC embedded into the player, and this NPC can be talked to by shift+c with nobody around. Sometimes the NPC would refuse to talk, or sass you. Sometimes it’d be nice. Treat the NPC poorly and your limbs could start rebelling against you, or you could start seeing shit. Treat them well and they can be your best post-apocalyptic imaginary friend.
Would be hilarious if they could give out certain missions, though I’m not sure if you’d get rewarded beyond relation increase.
Also would be silly if they’d have an in-game presence from time to time. Either just talking, or maybe an imaginary companion to help ward off the zombies, or punch you in the jaw. Maybe they attach to certain objects you carry around a lot, and you start drawing faces on them with a permanent marker?
Ol’ planky.