  The article summarizes and consolidates investigations on hyperbolic complex
numbers with respect to the Klein-Gordon equation for fermions and bosons. The
hyperbolic complex numbers are applied in the sense that complex extensions of
groups and algebras are performed not with the complex unit, but with the
product of complex and hyperbolic unit. The modified complexification is the
key ingredient for the theory. The Klein-Gordon equation is represented in this
framework in the form of the first invariant of the Poincar\'e group, the mass
operator, in order to emphasize its geometric origin. The possibility of new
interactions arising from hyperbolic complex gauge transformations is
discussed.
