We show that the set of complex numbers which are badly approximable by ratios of elements of $\mathbb{Z}[\sqrt{-D}]$, where $D \in \{1,2,3,5,7,11,19,43,67,163\}$ has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.