  The problem of the gauge hierarchy is brought up in a hypercomplex scheme for
a U(1) field theory; in such a scheme a compact gauge group is deformed through
a \gamma-parameter that varies along a non-compact internal direction,
transverse to the U(1) compact one, and thus an additional SO(1,1) gauge
symmetry is incorporated. This transverse direction can be understood as an
extra internal dimension, which will control the spontaneous symmetry
breakdown, and will allow us to establish a mass hierarchy. In this mechanism
there is no brane separation to be stabilized as in the braneworld paradigm,
however, a different kind of fine-tuning is needed in order to generate the
wished electroweak/Planck hierarchy. By analyzing the effective
self-interactions and mass terms of the theory, an interesting duality is
revealed between the real and hybrid parts of the effective potential. This
duality relates the weak and strong self-interaction regimes of the theory, due
to the fact that both mass terms and self-coupling constants appear as
one-parameter flows in \gamma. Additionally the \gamma-deformation will
establish a flow for the electromagnetic coupling that mimics the
renormalization group flow for the charge in QED.
