  Inspired by the appearance of split-complex structures in the dimensional
reduction of string theory, and in the theories emerging as byproducts, we
study the hyper-complex formulation of Abelian gauge field theories, by
incorporating a new complex unit to the usual complex one. The hypercomplex
version of the traditional Mexican hat potential associated with the $U(1)$
gauge field theory, corresponds to a {\it hybrid} potential with two real
components, and with $U(1)\times SO(1,1)$ as symmetry group. Each component
corresponds to a deformation of the hat potential, with the appearance of a new
degenerate vacuum. Hypercomplex electrodynamics will show novel properties,
such as the spontaneous symmetry breaking scenarios with running masses for the
vectorial and scalar Higgs fields, and the Aharonov-Bohm type strings defects
as exact solutions; these topological defects may be detected only by quantum
interference of charged particles through gauge invariant loop integrals. In a
particular limit, the {\it hyperbolic} electrodynamics does not admit
topological defects associated with continuous symmetries
