Linear fractional transformation fixing origin and preserving all distances

If you are considering only linear fractional transformations, then $0 \mapsto 0$ and $\infty \mapsto \infty$ (preserves all distances) imply $f(z)=az$. Since the transformation preserves all distances we have $a=e^{i\alpha}$, i.e. $f$ is a rotation.

But if you don't require $f$ to be a linear fractional transformation then let $e^{i\alpha}:=f(1)$. Since $f$ preserves all distances there are only two possibilities for each $re^{i\varphi}$: $$re^{i\varphi} \mapsto re^{i(\alpha+\varphi)} \quad \text{ or } \quad re^{i\varphi} \mapsto re^{i(\alpha-\varphi)}.$$

The first mapping is the rotation by $\alpha$, the second one is the rotation by $-\alpha$ followed by reflection in the real axis.