We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schrödinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. We use pseudo-arclength continuation to capture the relevant branches of solutions and explore their corresponding stability and dynamical properties (i.e., their fate when unstable). We focus on nine primary types of solutions: single site, double site in- and out-of-phase, squares with four sites in-phase and out-of phase in each of the vertical and horizontal directions, four sites out-of-phase arranged in a line horizontally, and two additional solutions having respectively six and eight nonzero sites. The chosen configurations are found to merge into four distinct bifurcation events. We unveil the nature of the bifurcation phenomena and identify the critical points associated with these states and also explore the consequences of the termination of the branches on the dynamical phenomenology of the model.