Are there any advantages of treating four-dimensional Euclidean space as $\mathbb{C}^2$ instead of $\mathbb{R}^4$?

Question

When we treat four-dimensional Euclidean space as $\mathbb{R}^4$ we can easily define a line, plane, and 3-plane by linear equations, whereas if we wish to do the same with $\mathbb{C}^2$ we have a harder time with lines and 3-planes. Are there any circumstances where using $\mathbb{C}^2$ is easier?