Well, to get non-real roots to a real polynomial, you extend the reals by adding a formal symbol, which is normally called 'i', but I will call 'h' just to be different, with the constraint that h^2=-1. So then you get some root involving h.
But now we say, "really" the root of -1 is 'i', so setting i=h gives the root we just found. But then we ask, what if 'i' is really -h? Could there be any difference? Since (-h)^2 is also -1 by elementary algebra, there is no way to say that h is "really" i or -i. But if a non-real value were a root without its complex conjugate being a root, then there would be a contradiction, because we would have to have a way of saying which of i and -i is "really" h.
This is an extremely informal argument, but I think the crux of the matter is that i is only defined in this formal manner.