Periodic points in complex continued fractions by J. Hurwitz

J. Hurwitz introduced an algorithm that generates a continued fraction expansion for complex numbers $\alpha \in \mathbb{C}$, where the partial quotients belong to $(1+i)\mathbb{Z}[i]$. J. Hurwitz's work also provides a result analogous to Lagrange's theorem on periodic continued fractions, describing purely periodic points using the dual continued fraction expansion. S. Tanaka \cite{ST} examined the identical algorithm and constructed the natural extension of the transformation generating the continued fraction expansion and established its ergodic properties. In this paper, we aim to describe J. Hurwitz's insights into purely periodic points explicitly through the natural extension.