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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?

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How do you like to visualize $\mathbb{R}^4$? – Elliott Jun 28 '11 at 22:47
I know that in complex analysis, to visualize a map $f: \Bbb{C} \to \Bbb{C}$ Riemann surfaces are used. They are hot easy to grasp though. Take a look at – Beni Bogosel Jun 28 '11 at 22:55
@Elliott: Are you claiming that these two vector spaces are isomorphic? – Fixee Jun 29 '11 at 3:14
They are isomorphic as vector spaces over $\mathbb{R}$ only. But no, I was trying to figure out how you visualize 4 spatial dimensions in the first place. – Elliott Jun 29 '11 at 9:01
Not sure if it is useful for OP's purposes, but anyway: – Ruben Jan 8 at 17:34
up vote 5 down vote accepted

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.

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