  An extension of the finite and infinite Lie groups properties of complex
numbers and functions of complex variable is proposed. This extension is
performed exploiting hypercomplex number systems that follow the elementary
algebra rules. In particular the functions of such systems satisfy a set of
partial differential equations that defines an infinite Lie group. Emphasis is
put on the functional transformations of a particular two-dimensional
hypercomplex number system, capable of maintaining the wave equation as
invariant and then the speed of light invariant too. These functional
transformations describe accelerated frames and can be considered as a
generalization of two dimensional Lorentz group of special relativity. As a
first application the relativistic hyperbolic motion is obtained.
