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There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested. In my opinion, the interesting geometry of the split-complex numbers cannot be reduced to the operations $\{+,-,\times,/\}$ over it, but rather you have to consider the involution $(a,b)^* := (b,a)$ as well. Maybe you can find some interesting phenomena there.

There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested. In my opinion, the interesting geometry of the split-complex numbers cannot be reduced to the operations $\{+,-,\times,/\}$ over it, but rather you have to consider the involution $(a,b)^* := (b,a)$ as well. Maybe you can find some interesting phenomena there. You might want to also check if operations on the extended real numbers can be combined with the involution to produce interesting phenomena.

There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested.

There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested. In my opinion, the interesting geometry of the split-complex numbers cannot be reduced to the operations $\{+,-,\times,/\}$ over it, but rather you have to consider the involution $(a,b)^* := (b,a)$ as well. Maybe you can find some interesting phenomena there.

There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested.