This 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 Poincaré group, the mass operator, in order to emphasize its geometric origin. The possibility of new interactions arising from hyperbolic complex gauge transformations is discussed.
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