Mean-field theory approximates many-body quantum dynamics by an effective single-particle evolution governed by nonlinear equations. As an introduction, I will present a quantum algorithm for solving nonlinear differential equations that relies on this mean-field approximation. Extending mean-field theory to non-Hermitian Hamiltonians could allow us to address more general classes of equations. The main part of the talk will focus on this non-Hermitian generalization, illustrating the associated challenges and potential approaches through concrete examples, and outlining the current progress of our ongoing work.