Starting from Maxwell’s theory of electromagnetism in a Minkowski spacetime, we generalize to arbitrary spacetimes and gauge groups. The gauge groups U(1) and SU(3) and their associated Yang-Mills theories are discussed in detail. 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 C2 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.