  We extend the work of An, Guan and Kleinbock on bounded orbits of
diagonalizable flows on $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$
to $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$, where
$\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a
complex analogue of Minkowski's Linear Forms Theorem. We then set up an
appropriate Schmidt game in $\mathbb{C}^3$ such that bounded orbits correspond
to a hyperplane-absolute-winning set consisting of certain vectors in
$\mathbb{C}^3$ relative to an approximation by imaginary quadratic rationals in
$\mathbb{K}$.
