  This paper provides a view of Maxwell's equations from the perspective of
complex variables. The study is made through complex differential forms and the
Hodge star operator in $\mathbb{C}^2$ with respect to the Euclidean and the
Minkowski metrics. It shows that holomorphic functions give rise to nontrivial
solutions, and the inner product between the electric and the magnetic fields
is considered in this case. Further, it obtains a simple necessary and
sufficient condition regarding harmonic solutions to the equations. In the end,
the paper gives an interpretation of the Lorenz gauge condition in terms of the
codifferential operator.
