Coquaternions were introduced in 1849 by the English mathematician James Cockle . The algebra of real coquaternions, also known in the literature as split-quaternions or paraquaternions, is an associative but noncommutative algebra over $\mathbb{R}$ defined as the set $$\mathbb{H}_{\rm coq} = \{q_0 + q_1{\mathbf i} + q_2{\mathbf j} + q_3{\mathbf k} : q_0, q_1, q_2, q_3 \in \mathbb{R}\}$$ where ${\mathbf i}$, ${\mathbf j}$, ${\mathbf k}$, called the imaginary units, satisfy $$ {\mathbf i}^2=-1, \ {\mathbf j}^2={\mathbf k}^2=1,\;\;{\mathbf i}{\mathbf j} = - {\mathbf j}{\mathbf i} = {\mathbf k}.$$
Coquaternions is a Mathematica add-on application whose main purpose is to define arithmetic for coquaternions. It adds rules to Plus, Minus, Times, NonCommutativeMultiply, etc and is still under construction.