$\mathbb{R}^n$ vs $\mathbb{C}^n$ geometric intuition

Question

I have good geometric understanding and intuition of $\mathbb{R}, \mathbb{R}^2$ and $\mathbb{R}^3$ and an abstract geometric intuition for higher dimensions, but I don't know how to geometrically think about $\mathbb{C}^2$, as a starter. Already $\mathbb{C}$ is imagined as $\mathbb{R}^2$, so where doees $\mathbb{C}^2$ and higher fit?

For example, when imagining orthogonal projections of vectors in $\mathbb{R}^2$, we can draw the vectors on paper and use the analogy of light casting a shadow onto the vector you are projecting to so that the shadow itself is the projection. How do you extend this to complex projections? What do you picture in your mind?

Answer 1

For now, it is reasonable to think of $\mathbb C^2 \cong \mathbb R^2 \times \mathbb R^2 \cong \mathbb R^4$, and more generally: $\mathbb C^n \cong \mathbb R^{2n}$.

A projection onto either co-ordinate, can be thought of as a projection $P:\mathbb R^4 \to \mathbb R^2$, if you already feel that this is helpful.

Answer 2

A geometric difference is that lines are thicker. You remove a point from them, and they don't become disconnected.

An algebraic difference is that linear transformations always have eigen-spaces. All of them have at least a direction in which they are multiplication by a scalar.

To notice that projections have the same geometric meaning, you can use the characterization of the orthogonal projection as the closest point of the subspace to the vector being projected. Then notice that the length defined by the standard complex inner product is the same as the length defined by the standard real inner product in the corresponding $\mathbb{R}^{2n}$.