Metrical properties of Hurwitz continued fractions

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.