Based on the analysis of biquaternion quadratic forms of field, it is shown that Maxwell equations arise as a consequence of the principle of conservation of the energy-momentum flow of field in space-time. It turns out that this principle presupposes the existence more general nonlinear field equations. Classical linear Maxwell equations are embedded in a special way into new nonlinear equations and are their special case. It is shown that in a number of important cases nonlinear equations, in contrast to linear ones, allow solutionsthat have a swirling energy flow. Solutions of the equations we obtained make it possible to give wave description of charged particles, common for quantum mechanics, within the framework of nonlinear classical electrodynamics. Special attention in the work is paid to the problem of dividing the field into "own" field of a charged particle and a field "external" to it. From the nonlinearfield equations follow both the classical Maxwell equations themselves and the equations of charges moving under the Lorentz force. In this way, the problem of finding nonlinear field equations that include interaction is solved. In our approach, the particle charge is electromagnetic (complex-valued), passingthrough periodically changing linear combinations of electric and magnetic charges - from purely electric to purely magnetic. In real processes, it is not the particle charge itself that plays a role, but its phase relationship with other charges and fields.