Almost certainly not. Although I have heard that everyone sees colours slightly differently, this difference will only be very slight (of course, there is no way to prove it, but it's a fair guess). Try looking at a brightly coloured object with one eye then the other and you might see a slight difference.
A good way to see why there can't be any new colours is to separate some white light out into a spectrum using a glass triangular prism. If you shine the separated light onto some paper then, as you know, red can be seen at one end, violet at the other, and orange, yellow etc. in between. However, this is not the complete spectrum. With typical sunlight, the separated bands of colour extend beyond what we can see: beyond red comes infrared; beyond violet comes ultraviolet. These 'invisible' frequencies of light are shining on the paper too, but we do not see them because our eyes can't detect them.
As for colours, such as brown and pink, which don't fit into the spectrum, I am guessing that they are an illusion caused by molecules, or areas, of differing colour being right next to each other and appearing to 'mix' into one colour. A bit like pixels on computer screens, maybe. White and red presumably combine in our eyes to appear as pink; a bit of everything jumbled together looks brown. There are animals that can see parts of the infrared or ultraviolet ranges of light, for example, snakes can find prey animals by detecting the infrared (heat) radiation that the animal emits. I've also been told that certain insects are attracted to flowers that are highly reflective to ultraviolet.