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