Can a person be born with an unusual hair color (red, blue, purple, pink)?

I remember a funny incident. In one well-known group, a young man of Caucasian nationality was indignant at the fact that boys now "have become like girls" - they follow fashion, dye their hair, use cosmetics, etc. Going to his page, I was amazed that a dude with bright red hair is so mad at dyed hair. Like he can, but others can't, or what? And then I learned that, it turns out, there are red-haired Caucasians (the template was not sickly). And not only are they red, some also have their hair color quite close to red. I have never met such shades. So the dude didn't suffer from double standards - red was his natural hair color. Which, of course, in no way justifies the system of his dense views.

In general, I answered one question - red hair really exists in nature. In vain, the answer about the red-red was minus, since he is absolutely right.
And the commentators above have already said everything else.

P.S. I will add a little about green hair. A person is not able to get such a shade from birth, but it is easy to earn it to an easy degree due to a certain composition of water. Brown hair turns green with a bang)

The bright colors of birds and insects, as already mentioned, are not caused simply by pigmentation. Since light during propagation behaves like a wave with a characteristic wavelength of half a micron, on objects of the corresponding order in size, wave effects significantly manifest themselves, in this case, interference in a thin film is like on a puddle into which gasoline has spilled. The thicker the film and the more the angle of view differs from the normal, the closer the color is to red, and the thinner it is to blue. For films that are too thin and too thick, the effect disappears.

In principle, a person can also develop some new pigment as a result of mutation, as once upon a time blue eyes appeared. The likelihood of such an event is extremely small, but a really bright pigment color is even less likely.

The repertoire of animal pigments (as opposed to plants) is very limited. As a first approximation, these are black-brown eumelanin and red phaeomelanin (everything else is rarely encountered, but never in humans). All blue and other interesting colors in bird feathers and on the wings of butterflies are formed by a certain spatial organization of pigments (how exactly the cells with the pigment are stacked relative to each other). Hypothetically, in order for a person to obtain blue hair in this way, it is necessary to rearrange the entire structure of the hair, and this requires many rearrangements in numerous genes. The probability of such a coincidence tends to zero.