Are we to assume that x is two dimensional?
I can somewhat picture this, but I'm having trouble with a number line for i (square root of negative one) with more than one dimension.
i is not a replacement for y is it? Or can Linear Algebra and Calculus be done with the Imaginary Number line standing in for y?
A video on Imaginary Numbers by Welch Labs on YouTube.
You should study Vector Spaces. This will formalize the concept of dimension. However, an important point is that a vector has an associated field. We usually say something like V is a vector space over F. Often the field is obvious and unstated but this is a case where you need to be fussy.
Triples of complex numbers could be regarded as a vector space over the field of complex numbers in which case the dimension would be $3$ not $6$.
The same set could also be regarded as a vector space over the field of real numbers in which case the dimension would be $6$.
It could even be regarded as a vector space over the field of rational numbers in which case the dimension would be infinite.
Any field is a vector space over itself and the dimension will always be $1$. The complex numbers are not inherently $2$ dimensional. They are $1$ dimensional over themselves. It is just intuition from day to day life and early mathematics which biases us towards the real viewpoint.