[This is a slightly tweaked re-posting of a question I asked in April 2020, that subsequently got auto-deleted as an "abandoned question." Hopefully launching a second attempt now (over two years after I first asked it, and I believe over a year after it was deleted) is okay.]
Tessarines are a product of sorts of the complex numbers and the split-complex numbers. If you substitute split-complex number coefficients for real coefficients in the form of complex numbers, or substitute complex number coefficients for real coefficients in the form of split-complex numbers, you get the tessarines. Complex numbers can be plotted in $R^2$ (two-dimensional Euclidean space with the normal metric, or the Cartesian plane), while split-complex numbers can be plotted in the pseudo-Euclidean space $R^{1,1}$. $R^{1,1}$ has a metric signature ($+$ $−$) or ($−$ $+$), and I assume that the real axis is generally associated with the $+$ and the "$j$-axis" (assuming the split complex number is written in the form $x+yj$ ) is generally associated with the $−$.
For tessarines (assuming they're of the form $w+xi+yj+zk$ (possible different variables than $w$, $x$, $y$ and $z$ or different order thereof aside) where $ij=ji=k$, $i^2=-1$, and $j^2=+1$ ), I'm fairly certain that they would have to be graphed on the pseudo-Euclidean space $R^{2,2}$, with two $+$s and two $−$s in the metric signature. I'm also fairly certain that the real axis and "$j$-axis" would have to have opposite signs (one $+$ and one $−$), and the same with the "$i$-axis" and "$k$-axis". (It would probably be more correct to label the axes after the variable representing the coefficients, but those variables and the order thereof might be less consistent among mathematicians than the use of $i$, $j$ and $k$ and their relationships to $1$ and $−1$ and to each other.)
I imagine, but am less certain here and my question is basically if my assumption is correct or not, that the real and $i$-axes are generally given have the same sign as each other (probably the $+$), and thus the $j$-and $k$-axes are also given the same sign (probably the $−$). While the $i$ and $k$ are equivalent to each other in mathematical operations involving tessarines, my assumption above would seem consistent with $i$ being the same as the $i$ in complex numbers and $j$ being the same as the $j$ in split-complex numbers, with $k$ being the product of $i$ and $j$.
As axes in non-positive definite coordinate spaces are sometimes thought of as representing either space or time dimensions, you could think of the real axis as representing a real space dimension, the $i$-axis as representing an imaginary space dimension (imaginary in the same sense as in the complex numbers), the $j$ axis as representing a real time dimension and the $k$-axis as an imaginary time dimension. While I have talked above about tessarines being plotted in $R^{2,2}$, it occurs to me that this space of four real dimensions might be more appropriately called $C^{1,1}$, although I don't know if it's kosher to use an indefinite signature with a "base" algebra other than the real numbers to describe a coordinate space.
I could be wrong, however, about there being a standard for how tessarines are plotted on $R^{2,2}$, and maybe plotting tessarines in multi-dimensional space in a way analogous to complex-numbers (the complex plane), quaternions and split-complex numbers just isn't a thing. There are special names for pseudo-Riemannian manifolds with $0$ (Riemannian) or $1$ (Lorentzian) dimensions with the opposite "sign" from all other dimensions, but there doesn't seem to be much attention paid to pseudo-Riemannian manifolds or even pseudo-Euclidean spaces with two or more dimensions of each "sign."
I can't be the only person who's thought of this, however. I'd appreciate whatever help in answering my "question" and confirming or correcting my assumptions that people can give.
Best Answer
In tessarine modulus (which is not a norm as it does not satisfy the norm axioms) real and the two imaginary axes have positive sign before square under root and the hyperbolic axis has negative sign. So, 3 positive and one negative.
In other words,
$|a+bi+cj+dij|=\sqrt{a^2+b^2-c^2+d^2}$.
Hyperbolic axes get negative sign, real and imaginary get positive sign, dual (if any) have zero coefficient (do not contribute to modulus).