Does complex axes exists? How such a coordinate systems are called?

Could a coordinate on a coordinate axis (of some unusual coordinate system) be a complex numbers itself?

For example, if I consider some 2D coordinate system, where x and y both being a complex numbers, how could I define euclidean distance? I want to test - how would inverse square-typed laws work there (and even could they exist?), for example newton gravity: $$F=G m_1 m_2 \vec{r12}^{-1}$$ where term r12 inverse is usually done like $$\vec{r12}^{-1} = \frac{\vec{r12}}{|\vec{r12}|^2}$$

r12(=r2-r1) here is a 2 elements vector, with each element being a complex number, expanded like: $$\vec{r12_x} = (\vec{r2_{x_{re}}}-\vec{r1_{x_{re}}}, \vec{r2_{x_{im}}}-\vec{r1_{x_{im}}})$$, and the same way for y coordinate

How should I ideologically do? Is euclidean distance for inverse term calculation actually a $$|\vec{r12}|^2$$? or a $$\vec{|r12^2}|$$? Does it even exist? I've messed up.. Is that system fully equal to 4-dimensional (quaternion math)? or no?

It feels like a norm squared is a complex number itself, and a r12 dual complex, we should take each part x and y and divide it by squared norm, according to complex division

• I'm no expert, but having complex numbers as the "markings" on 2D axes makes no sense since you can't really order complex numbers. But you should look up Quaternions, which might fit your bill. They make up a four dimensional system and are a generalisation of complex numbers. – Deepak May 2 at 15:55
• @Deepak quaternions or tessarines(bicomplex)? For example i want to make mass an complex number, so force on such a 2d plane would become 4dimensional.. – xakepp35 May 2 at 16:01
• I don't understand the goal you are outlining, but "having complex coordinates" and "euclidean distance" sounds an awful lot like ordinary complex vector space theory. The distance I'm suggesting is the usual complex conjugate form. – rschwieb May 2 at 16:48
• @rschwieb I thought like what if some simple things, like mass, or charge are complex numbers or even quaternions?. It is not so in our world, so I am trying to define some alternative world(with new axiom and laws system), in which it is so. Taken this as some axiom (no "why", just "if"), I would like to check if such a world could exist "on a paper" - e.g. should Couloumb's, Ohm's, Watt's et.al laws satisfy, and what should be changed. Obviously, typical coordinate systems (like common 2d or 3d vectors) is something that could not be used and should be changed in a first place. – xakepp35 May 2 at 17:18
• @xakepp35 You can do linear algebra with any division ring, and Desargues holds in any dimension of geometry. What you lose is things like length and angle. And if you're using fields of positive characteristic, there isn't really any notion of distance at all. – rschwieb May 2 at 17:24