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In this contribution we sketch a branch-cut quantum formulation of the Wheeler-DeWitt equation analytically continued to the complex plane. As a starting point, we base our approach on the Hořava-Lifshitz formulation of gravity, which employs higher spatial-derivative terms of the spacetime curvature for renormalisation reasons. Following standard procedures, the quantization of the Lagrangian density is achieved by raising the Hamiltonian, the dynamical variable which represents the branch-cut complex scale factor and the conjugate momentum to the category of operators. We arrive at an Schrödinger-type equation with a non-linear potential. Solutions are then obtained and discussed for different potential parameterizations. The results reinforce the conception of a quantum leap between the contraction and expansion phases of the branch-cut universe, in good agreement with the Bekenstein criterion.