This may be a stupid question.
I am trying to explain/prove the following:
Let $f$ be a complex rational function of degree $d ≥ 2$ with fix point $z_0 \neq 0$. It is always possible to conjugate $f$ such that the conjugated $\overline{f}$ has a fix point $z_0 = 0$.
My intuition is that you can use an involution $\gamma$ to conjugate $f$ as follows $$\overline{f} = \gamma \circ f \circ y$$
So my question is, how can I be sure that it is always possible to conjugate $f$ such that the fix points are sent to zero?