The impedance matching control, also known as approximate complex conjugate control (ACC), is one of the main strategies for improving the capture of energy by point absorber wave energy converters. Such a strategy shapes the mechanical impedance related to the floater dynamics via the control law. Since the traditional ACC is given by a linear control law, this work proposes a generalization denoted as nonlinear complex conjugate control (NCC) that considers the presence of nonlinear viscous damping in addition to the usual linear damping and stiffness. The energy maximization conditions for the proposed NCC are derived in the frequency domain through the describing function method. These conditions show that the ACC is a special case of the NCC when the total damping on the floater is approximated as a linear function of its velocity. From numerical simulations of a point absorber wave energy converters with nonlinear damping, which is based on the Archimedes wave swing prototype, it is shown that the NCC provides greater energy conversion than the ACC, as well as a robust performance in the presence of variations of the damping coefficient and the excitation force peak frequency.