There is a thread on Reddit:

Quote Originally Posted by DR_Trukey
I'm not sure if this is an idea that's been brought up before. I'd like to see something like wigs in the game. Basically you'd make a hairstyle in the character creation screen like normal and that becomes a piece of race/gender specific head gear that you can glamour so you can have different hair styles with different equipment sets.

I don't know how hard it would be to implement on the programming side but changing your hair only costs 2k gil in game so it's not like SE would be losing money on Fantasias by incorporating it.

Has this idea been brought up before?
I thought it should be suggested here too. The specifics of how o do it might vary, but just being able to get a wig with a hairstyle, and dye it, and then glamour it onto a headpiece, would be fun.