Complex powers $z^\alpha$ along a branch of the logarithm

Let \(\Omega\subseteq\CC\setminus\{0\}\) be open. A branch of the logarithm on \(\Omega\) is a continuous function \(L\colon\Omega\to\CC\) with \(e^{L(z)}=z\) for all \(z\in\Omega\). For such \(L\) and \(\alpha\in\CC\), the branch of \(z^\alpha\) determined by \(L\) is \[ z^\alpha\coloneqq e^{\alpha L(z)},\qquad z\in\Omega. \]