Split-quaternion like extension of the complex plane

This paper introduces an inspiring set extension for the complex plane. The reason behind it is to discuss a transformation that mimics Lorentz transformations and defines a tensor with both upper and lower index representations of four-position. This will allow to define the necessary metric that raises and lowers the tensor index, as well as the inner product that leads to a Lorentz scaling function, showing how the model offers a new homogeneous isotropic universe, that has a well-defined cosmological density and pressure, while at the same time being consistent with special relativity. Given assumptions on the evolution of the parameters that define the model, a scaling factor curve that is qualitatively close to the standard Lambda-CDM one will be found, without invoking any extra knowledge of the cosmological constant, or even the matter content of the model. The paper is purely motivated by mathematical concerns.