TL;DR: how do I define a "split-holomorphic" function?
As far as I've heard, there is a notion of split-complex numbers: $z = x+uy$, with $u\not\in \Bbb R $ and $u^2 = 1$. One defines the conjugate as $\overline{z} = x - uy$ and the modulus as $|z| = \sqrt{|z\overline{z}|}$, where the absolute value inside the root is the one for real numbers, as usual. Apparently this can be used to study the pseudo-Riemannian geometry of $\Bbb L^2 = \Bbb R^2_1$ in the same way thay $\Bbb C$ is identified with $\Bbb R^2$.
But these split-complex numbers form a ring only, and not a field: elements in the diagonals don't have multiplicative inverses. Incidentally, these diagonals correspond to lightlike directions in $\Bbb L^2$ (and such directions cause every sort of problems).
I don't know if there is a standard notation for this set of split-complex numbers, but given a function of said set to itself, how do I define things like "being holomorphic"? I can't make sense of the limit $$\lim_{h\to 0} \frac{f(z_0+h)-f(z_0)}{h}$$ since $h$ could approach zero from lightlike directions. Do I just ignore these directions? Most likely I'm overthinking this.
We could define continuity of $f$ by continuity of its components, and work formally with the above limit to obtain a revised version of the CR equations, and say that $f$ is split-holomorphic if the revised CR equations hold and if the partial derivatives of the components are continuous (copying the Looman-Menchoff theorem), but I am speculating too much and I am unsure about these things.
What is the correct way to do calculus with split-complex numbers, if this is possible? More references are also welcome.
P.s.: there is a nice book called "Geometry of Minkowski Space-Time", by Zampetti, et al, but they do a more algebraic approach and do not talk about the things I'm asking.
