I've have defined below new set of like-complex numbers (will be called $\Bbb{HC}$ here as my proposition for name for them is hexagonal-complex):
Let $z = (u,d) \in \mathbb{HC}$ where $u,d \in \mathbb{R}$ such that:
- $(1,0)(1,0) \triangleq (0,-1)$
- $(0,1)(0,1) \triangleq (-1,0)$
- $(0,1)(1,0) \triangleq (1,0)(0,1) \triangleq (0,1)+(1,0)$
- $\mathbb{HC}$ is a 2d vector space over $\mathbb{R}$
Such defined numbers have following properties, which you can check for yourself:
- $z_1 z_2 = z_2 z_1$
- $z_1(z_2+z_3) = z_1 z_2+z_1 z_3$
- $(1,1)z_1 = z_1$
If we define
- $z^* = (u,d)^* \triangleq (d,u)$
- $|z| \triangleq \sqrt{zz^*}$
Than one can easly proove that: $|z_1 z_2| = |z_1| |z_2|$
My name hexagonal-complex is from the fact that you can do an identification to complex numbers as:
$(1,0) \leftrightarrow e^{i \pi /3}$
$(0,1) \leftrightarrow e^{-i \pi /3}$
And the lattice of points $(u,d)$ where $u,d \in \mathbb{Z}$ is a hexagonal latice in complex plane.
And finally the question is: do such defined numbers have a name other than what I called them?
$\Bbb{HC}$ does not define a hexagonal lattice in the complex plane; $(1,1)$ is mapped to 1 using your $e^{i\pi/3}$ and $e^{-i\pi/3}$ mappings, which would lie in the centre of one of the hexagons you perceive. Indeed, it's easy to see that a triangular lattice is formed instead, and your set of integral points in $\Bbb{HC}$ are the Eisenstein integers.
Since $\Bbb{HC}$ is a 2-dimensional vector space like $\Bbb C$, it is merely $\Bbb C$ with a linear transformation, not anything new.