Would it be possible to invent some new color, for example "purple", so that identifying with that color would increase someone's IQ?

It's been done - many people do in fact self-identify as 'Indigo children', 'Indigos' or even 'Brights'. The label tends to come with a broadly humanistic and strongly irreligious worldview, but many of them are in fact highly committed to some form of spirituality and mysticism: indeed, they credit these perhaps unusual convictions for their increased intelligence and, more broadly, their highly developed intuition.