 Möbius conjugacy normal forms

A nonidentity Möbius transformation has one or two distinct fixed points on the Riemann sphere. If there are two, send them to zero and infinity; the conjugated map fixes both and is a nonzero scaling. If there is one, send it to infinity; the resulting affine map cannot have slope different from one, since that would give another fixed point. It is a nonzero translation, which a scaling conjugates to $z+1$. The identity is the scaling $\mu =1$. Fixed-point sets are carried bijectively under conjugacy, so a translation is not conjugate to a scaling.