There are two issues associated with "seeing extra colors". One is seeing parts of the light spectrum we currently do not see; for example, seeing ultraviolet or infrared light. Another is discriminating between different combinations of wavelengths, and perceiving more colors than we actually do.
Our color vision is based on three different receptors who are sensitive to different wavelengths; the brain infers color from the differences between how those receptors are activated. This allows to tell some combinations of wavelengths from others, but not all: we can tell red light from blue light, because the first excites the red-detecting cells a lot more than the two others and the second excites the blue-detecting cells a lot more than the two others, but we cannot tell yellow light from a combination of red and green light, because in both of those cases we have the red and green cells excited the same and the blue cells not as much. This is why our color vision is "three-dimensional", i.e. all colors can be generated from three basic ones.
Somebody who instead of a blue receptor had an ultraviolet receptor, would be able to perceive more wavelengths than we do, but they would probably see more or less the same colors - "ultraviolet" would be "blue" to them. There may be differences because the kind of combinations they ran into would be different - for example because they'd have a wider range between the "green" and "ultraviolet" sensors than we have between "green" and "blue", maybe more of the world would look cyan than it does to us.
To see additional colors, the same way that ordinary trichromat humans see more colors than color-blind people, and be able to discriminate between more combinations of wavelengths than we do now, we would need extra receptors. And that's where the questions arise of whether the brain would be able to handle the extra colors, or whether the optic nerve would transmit it as its own channel.
As it happens there are cases in humans where they have four different color receptors. The main case happens because the genes for some color receptors (red/green) are on the X chromosome, and in women one of their two X chromosome is inactivated in each cell, but which it is is random - meaning that if they have two slightly different versions of that receptor on their two chromosomes, some cells in the retina will have one and others the other - making two different color receptors in their eyes where other people have one.
It appears that most women with this conditions do not see more colors than others, which could be because the two receptors don't vary enough. But it appears that some rare cases do have better abilities to discriminate colors and a richer color experience, which suggests that human tetrachromacy is possible and already exists in a few rare individuals.
Enhancing human color vision by breaking binocular redundancy (this isn't about the abovementioned cases of tetrachromacy, but of people trying to induce tetrachromacy by stimulating the two eyes differently)
This paper gives a good general overview of color vision in the animal kingdom, and debunks some misconceptions (such that animals with a zillion color receptors see a bazillion more colors than we do - they actually process color very differently and the number of color receptors they have is related to that).
Another overview of human color vision:
Human Potential for Tetrachromacy