I was under the impression that the Mobius maps were exactly all of the bijections $f:\mathbb{C_{\infty}} \mapsto \mathbb{C_{\infty}}$, however I came across the following question
Show that an invertible function $f:\mathbb{C_{\infty}}\mapsto \mathbb{C_{\infty}}$ that preserves the cross ratio is a Mobius transformation.
which clearly seems to imply that this is not the case! Clearly all maps of the form $z \mapsto \frac{az+b}{cz+d}, ad-bc\neq 0$ are bijections. But I cannot think of a bijection $\mathbb{C_{\infty}}\mapsto \mathbb{C_{\infty}}$ that is not of this form and thus would not be a Mobius map! Any help would be much appreciated!