Biholomorphisms are conformal and have holomorphic inverse

A biholomorphism is holomorphic and has a holomorphic inverse by Biholomorphic maps between complex domains, so it is conformal in the orientation-preserving sense this page uses: holomorphic with nowhere-vanishing derivative. The derivative cannot vanish at any point of its domain, because the local inverse supplied by Holomorphic inverse function theorem and local-degree criterion has derivative $g^{\mathrm{\prime }}(w)=1\mathrm{/}{f}^{\mathrm{\prime }}(g(w))$; a vanishing $f^{\mathrm{\prime }}$ would make that expression undefined. Such a map preserves the magnitude and the orientation of angles between tangent directions at every point.

The convention here is deliberately orientation-sensitive: complex conjugation $z\mapsto \bar{z}$ preserves angle magnitudes but reverses orientation, so it is not conformal in this library's sense. That exclusion is exercised by the companion page's conjugation counterexample.