  We develop the geometry of Hurwitz continued fractions -- a major tool in
understanding the approximation properties of complex numbers by ratios of
Gaussian integers. We obtain a detailed description of the shift space
associated with Hurwitz continued fractions and, as a consequence, we
contribute significantly in establishing the metrical theory of Hurwitz
continued fractions, analogous to the well-established theory of regular
continued fractions for real numbers.
  Let $\Phi:\mathbb N\to \mathbb R_{>0}$ be any function and $a_n(z)$ denote
the $n$th partial quotient in the Hurwitz continued fraction of a complex
number $z$. The main result of the paper is the Hausdorff dimension analysis of
the set \[E(\Phi) \colon= \left\{ z\in \mathbb C: |a_n(z)|\geq \Phi(n) \text{
for infinitely many }n\in\mathbb{N} \right\}. \] This study is the complex
analogue of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5,
1319--1339].
