Interpretation of Complex numbers - confused

[closet mathemetician]

I know that generally complex numbers are represented in a two-dimensional plane with one real and one imaginary dimension. I also know that we have the quaternions, consisting of one real number and three imaginary numbers.

The imaginary axis is always perpendicular to the real axis. The problem is that in our world we have three mutually orthogonal real dimensions. So if the imaginary dimension has to be perpendicular to all three real dimensions, that would make the imaginary dimension a fourth dimension.

Now we have the problem that there are three different planes with a real dimension and an imaginary dimension so we have three complex planes. It seems to me that you would have a four-dimensional vector, with three real components and one imaginary component:

V=(x,y,z,it)

using "t" as the fourth dimensional real number by which to multiply the imaginary unit.


The "regular" inner product of V with itself would give the Minkowski metric

V^2=(x^2+y^2+z^2-t^2)


I have never seen the imaginary dimension presented this way. Why not? Can anyone explain?

 

[closet mathemetician]

Quoting myself,

Now we have the problem that there are three different planes with a real dimension and an imaginary dimension so we have three complex planes.

Could this be the basis for quaternions, i.e., three complex planes? However, there is really only one imaginary dimension involved with all three planes.