  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
solutions that 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 nonlinear field 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), passing through 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.
