Why is my coworker always wearing a hat?

She knows and we do not. If you've never seen her hair, a religious requirement or medical issue are both reasonable guesses. Another possibility is that her hair is some bright artificial color (or colors) that suit her during off duty hours but don't give the professional look she needs for her job. 

Most likely you have no need to know and shouldn't ask her. If the two of you become friends, the explanation might come up naturally sometime. 

she might be one, maybe you should ask her

It doesn’t matter and it’s none of your business. 

Maybe it's a medical issue she chooses to keep private, like alopecia, or cancer.

It could be a religious requirement or maybe she prefers wearing a hat instead of the usual head coverings.

I know an Egyptian woman who does that and she is a Muslim. She could be, just ask her. 

They feel naked without it

I don’t know he must be weird hon