Contractions which are conformal

Question

Can someone give an example for a contraction which is also a conformal? I know that we call a contraction as conformal when it preserves angles. Even a hint would be helpful.

Answer 1

Unless I am misunderstanding your request, $z\mapsto\frac12z$ is both a contraction and a conformal map from $\mathbb C$ into $\mathbb C$.