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I don't think a colour of hair have anything to do with professionalism.
Choice of clothes, accessories, or make-up, sure. Hair colour? Not really.

It depends what profession you're in. But bright pink hair isn't typically associated with being business oriented. Some people want their employees to be presentable to their clients and bright pink hair can be seen as not appropriate in their eyes. It's not a matter of being oppressed, just that the company doesn't want someone in their organization to present themselves that way.

You could make the same argument against people with blonde hair. (Because you know, blondes are stereotypically, well, dumb)

So it's the same argument against someone whose genetics made their hair a specific vs someone who intentionally made it bright pink by their choice? No I'd say a more similar scenario would be a guy who tattooed his face and doesn't understand why he is told he looks unprofessional.

Right, because a female, or a male have never dyed their hair from red to blonde, or from brown to blonede, or whatever, eveyone always keep their natural "genetic" colour, unless they dye it pink...
It is very same, if you're referring to what's stereotypically associated with professionalism, if you buy into stereotypes about bright, atypical hair colours, then you might aswell buy into stereotypes about bright typical hair colours.
Neither one of them actually have influence on individual professionalism.
As for facial tattoos, it's not about professionalism but about the intimidating look, the same you can say about scars, which are not a choice, although there are idiots who make themselves facial scarifications. Unless we're talking about working with clients face to face, I don't think an employer would care less.

I did say it depends on the profession. Whether you think it's true or not, there are people who don't feel it comes off as professional. Companies don't want to associate themselves with people who don't fit with their brand. Of course these cosmetic changes don't indicate the person's actual professional behavior. However we live in a superficial cosmetic world where appearances are important to people. I personally don't see anything wrong with dying your hair a color that is not natural but some employers don't want their brand associated with it. (and by natural I meant colors that you would never see on someone's head without a dye job, changing from one natural color to another isn't the same thing as making your hair glow under a black light) Feminists are very common to have their hair dyed un-natural colors (once again not their personal natural hair color, but a color that is not found in nature) and feminists are always claiming that they are oppressed. So it commented that many companies would see bright pink hair has not presentable to the image they want associated with their brand. And yes I was referring to jobs in which they are interacting with clients. But certain companies have specific dress codes, which can be ridiculous in themselves. However the employees are representatives of the company. So you said with facial tattoos it's about the intimidating look, not professionalism. Then you really should define professionalism. Because while the cosmetic aspects doesn't dictate the actions of the individual while working, if it doesn't present the image the company wants to portray (for example a high end classy brand having a person with green hair attempting to sell their product.) has an effect on their work (such as client face to face work) then it's not being professional. Some employers would not care, but others don't feel it suits their company image.