The following is not well-known, but (modified) Maxwell equations can indeed describe both electromagnetic field and electrons.
@Quantumwhisp commented: "Maxwell's equations don't describe charged particles at all", and then asked: "Can you derive the Lorentz-Force from maxwell's equations?"
I am not saying these comments are unreasonable, but, surprisingly, Dirac did derive the Lorentz force from Maxwell equations (Proc. Roy. Soc. London A 209, 291 (1951)).
I summarized Dirac's derivation elsewhere as follows.
Dirac considers the following conditions of stationary action for the free electromagnetic field Lagrangian subject to the constraint $A_\mu A^\mu=k^2$:
\begin{equation}\label{eq:pr1}
\Box A_\mu-A^\nu_{,\nu\mu}=\lambda A_\mu,
\end{equation}
where $A^\mu$ is the potential of the electromagnetic field, and $\lambda$ is a Lagrange multiplier. The constraint represents a nonlinear gauge condition. One can assume that the conserved current in the right-hand side of the equation is created by particles of mass $m$, charge $e$, and momentum (not generalized momentum!) $p^\mu=\zeta A^\mu$, where $\zeta$ is a constant. If these particles move in accordance with the Lorentz equations
\begin{equation}\label{eq:pr2}
\frac{dp^\mu}{d\tau}=\frac{e}{m}F^{\mu\nu}p_\nu,
\end{equation}
where $F^{\mu\nu}=A^{\nu,\mu}-A^{\mu,\nu}$ is the electromagnetic field, and $\tau$ is the proper time of the particle ($(d\tau)^2=dx^\mu dx_\mu$), then
\begin{equation}\label{eq:pr3}
\frac{dp^\mu}{d\tau}=p^{\mu,\nu}\frac{dx_\nu}{d\tau}=\frac{1}{m}p_\nu p^{\mu,\nu}=\frac{\zeta^2}{m}A_\nu A^{\mu,\nu}.
\end{equation}
Due to the constraint, $A_\nu A^{\nu,\mu}=0$, so
\begin{equation}\label{eq:pr4}
A_\nu A^{\mu,\nu}=-A_\nu F^{\mu\nu}=-\frac{1}{\zeta}F^{\mu\nu}p_\nu.
\end{equation}
Therefore, the last three equations are consistent if $\zeta=-e$, and then $p_\mu p^\mu=m^2$ implies $k^2=\frac{m^2}{e^2}$ (so far the discussion is limited to the case $-e A^0=p^0>0$).
Thus, the first equation with the gauge condition
\begin{equation}\label{eq:pr5}
A_\mu A^\mu=\frac{m^2}{e^2}
\end{equation}
describes both independent dynamics of electromagnetic field and consistent motion of charged particles in accordance with the Lorentz equations. The words "independent dynamics" mean the following: if values of the spatial components $A^i$ of the potential ($i=1,2,3$) and their first derivatives with respect to $x^0$, $\dot{A}^i$, are known in the entire space at some moment in time ($x^0=const$), then $A^0$, $\dot{A}^0$ may be eliminated using the gauge condition, $\lambda$ may be eliminated using the first equation for $\mu=0$ (the equation does not contain second derivatives with respect to $x^0$ for $\mu=0$), and the second derivatives with respect to $x^0$, $\ddot{A}^i$, may be determined from the first equation for $\mu=1,2,3$.
However, the above is about classical electrodynamics. What about quantum theory? It turns out that modified Maxwell equations can be equivalent to the Klein-Gordon-Maxwell electrodynamics or (with some caveats) to the Dirac-Maxwell electrodynamics (see my article Eur. Phys. J. C (2013) 73:2371 at https://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4 ).
