I ran into an interesting result which showed that if $A$ is an orthogonal matrix then $$\left\vert\left\vert Ax \right\vert\right\vert = \left\vert\left\vert x \right\vert\right\vert$$ where $x \in \mathbb{R}^n$.
So now I am wondering if all orthogonal mappings are also conformal mappings.