The book I'm reading through stated the following:
Let $\mathbb P^1$ be the complex projective line, pick a random line that doesn't pass through the origin in $\mathbb C^2$. Then any point $p\in\mathbb P^1$ (i.e. a line through origin) will intersect the affine line at exactly one unique point, except the parallel line, which we denote as the $\infty$.
I guess my question is about how to visualize complex vector spaces. I see how the above would work for $\mathbb R$. Two lines will intersect at exactly a point. But for the case of $\mathbb C$, while I can tell why two complex lines intersect at a point by linear algebra, my doubt is both the one-dimensional subspace and the affine line basically "look like planes", and so their intersection should "look like a line". My question is how does this correspond to a point? My speculation is that thinking $\mathbb C^2$ as something like $\mathbb R^4$ is the wrong way to visualize it.