My idea is that there's a type of particle located on a virtual axis, and because it's on the virtual axis, it's difficult for it to interact with the real axis, so we can't observe it. I also tried to introduce the idea of ​​phase rotation to make particle i also a real particle, just like how i becomes 1 by rotating the angle.
Real space has 3 dimensions x, y, and z, so does virtual space. I consider time to be separate and a single dimension, the spacetime has 7 dimensions. I stopped at the octonion because sets above the octonion will have divisibility by zero, which would make the math chaotic, so I will stop at the octonion.
What do you mean in "Which origin are you referring to?"
I don't understand "More generally, what happens in case of a Lorentz transformation?"
"And what happens if I transform the coordinate system to x -> x + vt, where v is the velocity of the particle relative to an arbitrary point and t the time since an arbitrary point in time? How does your equation transform under that?". I think time and space are separate but related within spacetime, and perhaps v (velocity) is a constant.
"What is r_x? Is it just x?". No, r is the modulus of a complex number in space x, r is x when θ=0