Let a unitary number be one that corresponds to a matrix of the form: $$\left( \begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x) \end{array} \right)$$
This is analogous to quaternions where $p=2\pi n | n\in\mathbb{Z}$.
So I have two unitary numbers $(Z|C)\in\mathbb{U\left(2\right)}\times\mathbb{R}^+_0$ with $Z=\left(w,x,y,z,p\right)$ and $C=\left(a,b,c,d,q\right)$ with $\left(a|b|c|d|p|q|w|x|y|z\right)\in\mathbb{R}$.
Defining multiplication is very straight forward and so is defining powers or inversions. - This is, after all, just a unitary matrix that is not noramlized.
Is there, however, a "right" way to define addition, analogous to the addition of quaternions?
For $p=q$, this is straight forward and can be done just like for quaternions. However, what about the case, where $p\neq q$ ? Is there a sensible way to do this?
I.e. $Z+C = \left(w+a,x+b,y+c,z+d,\phi\right)$ - if it is at all possible in a natural way, how do I calculate that $\phi$?
I already tried $\phi=\arg\left(r e^{i p}+s e^{i q}\right)$ with $r=\sqrt{w^2+x^2+y^2+z^2}$ and $s=\sqrt{a^2+b^2+c^2+d^2}$, which would be analogous to complex numbers but the phases don't exactly correspond to the polar components of these unitary numbers (which would be $\frac{Z}{r}$ and $\frac{C}{s}$, respectively), so this definition is rather arbitrary.
I'm unsure about the tags. Please retag if necessary.