Ten years ago I started GMing a 3rd edition D&D game set in the Forgotten Realms. One of my players had the crazy idea of playing this guy called "The General" who had been turned undead by a lich and could now remove his head from his body and use it to spy on people or search around corners. In truth, he confessed, the character was the lich but had been driven insane from his many years of lichdom and now was coping by denying that he himself was the lich. "The General" was just a fiction he had created for himself, or maybe it was somebody who had served him at one time but was now long dead and in his madness had adopted the identity.
Back then, I allowed the character.
I created secret caveats though. The player was already playing with madness and I used that to mess with him. I made the character a secret psionicist who didn't know how to cope with his powers, and rather than being an undead creature who could detach his own head, the General carried around a sock puppet head with buttons sewn on that he believed was his detachable head. At some point I retconned that he telekinetically flew around the sock puppet head and used clairvoyance/clairaudience upon it. But the first time he threw that head the other players were confused as hell, I let the player describe what the General was doing then said "Okay, that's what you believe you're doing" then turned to the other players and described what the other characters actually saw him do. It was a hilarious moment.
Today, I would still allow the character but would create no secret caveats.
The character would simply be as the player described, unless he gave me free reign to muck about with his backstory. But more than likely, I wouldn't. It's a unique and novel idea, and I rarely encounter players who have highly detailed ideas about what they want their characters to be. There are plenty of people who just play straight up classes with few eccentricities or no uniqueness, but it's a real treat to find a player who wants their character to have depth and a layered history before they even start playing. It gives reasons right away for a GM to engage that character into a story, and easy hooks to pull on.
This is a rule that every GM should live by: If a player comes to you with a character concept that they really want to play, let them play it.
Now I'm thinking about all of the times I had some rules-lawyering or campaign-minded GM who didn't let me play some unique character I had come up with, or found a way to cripple my concept within a session or two.
I don't play with any of those assholes anymore.