  We investigate the non-Hermitian Haldane model on hyperbolic $\{8, 3\}$ and
$\{12, 3\}$ lattices, and showcase its intriguing topological properties in the
simultaneous presence of non-Hermitian effect and hyperbolic geometry. From
bulk descriptions of the system, we calculate the real space non-Hermitian
Chern numbers by generalizing the method from its Hermitian counterpart and
present corresponding phase diagram of the model. For boundaries, we find that
skin-topological modes appear in the range of the bulk energy gap under certain
boundary conditions, which can be explained by an effective one-dimensional
zigzag chain model mapped from hyperbolic lattice boundary. Remarkably, these
skin-topological modes are localized at specific corners of the boundary,
constituting a hybrid higher-order skin-topological effect on hyperbolic
lattices.
