  In this paper we present a new geometric formulation (Clifford algebra
formalism) of the field equations, which is independent of the reference frame
and of the chosen system of coordinates in it. This formulation deals with the
complex 1-vector $\Psi =E-icB$ ($i$ is the unit imaginary), which is
four-dimensional (4D) geometric generalization of Majorana's complex 3D
quantity $\mathbf{\Psi}=\mathbf{E}-ic\mathbf{B}$. When the sources are absent
the field equations with the complex $\Psi $ become Dirac-like relativistic
wave equations for the free photon. In the frame of ``fiducial'' observers (the
observers who measure fields are at rest) and in the standard basis the
component form of the field equations with 4D $\Psi $ reproduces the component
form of Majorana-Maxwell equations with 3D field $\mathbf{\Psi}$. The important
differences between the approach with the 4D $\Psi $ and that one with the 3D
$\mathbf{\Psi}$ are discussed.
