Note that if $a$ is part of a pair that is conjugate (we shall also call it conjugate), it is a root of $f(f(x)) - x$ which is quartic. However this polynomial also has a factor of $f(x) - x$, which is quadratic, so the polynomial $f(f(x)) - x$ decomposes into two quadratic factors one of which is for the fixed points of $f$ and the other is for the conjugate numbers. However, if $a$ is conjugate, then $f(a)$ is also conjugate, and so there are no other conjugate numbers; thusly, if any conjugate numbers exist there must be exactly two. But we know of two conjugate numbers ($a$ and $b$), so there can be no more, and we are done.