Is there a "good" way to visualize complex vectors?

We often represent complex numbers as vectors in $\mathbb{R}^2$ with $x$ being the real axis and $y$ being the imaginary axis. We often represent 2-dimensional vectors over $\mathbb{R}$ in a similar way.

Suppose we consider $\mathbb{C}^2$, vectors in two dimensions over $\mathbb{C}$. It feels like the complex plane is "embedded" into the scalars and I would like to somehow visualize these planes in the context of $\mathbb{C}^2$.

Is there a "good" way to think about this that people find intuitive?

Solution 1:

The two "complex axes" might be visualized as a pair of planes that intersect at a point rather than a line.

Incidentally, the first chapter of Kendig's Elementary Algebraic Geometry is devoted to helping visualize hypersurfaces in $\mathbb C^2$. It has some really great drawings and figures that give a concrete sense of the topology of various algebraic varieties.