I have seen this question and detailed answers about complex hypersurfaces here. However, I want to ask how to visualize an affine hypersurface.
For example, in $\mathbb{C}^2$, if $H$ is a complex line that passes through $0$, then I know that $H$ is the real span of $\{v, iv\}$ for some $v\in H$. This is indeed a plane in $\mathbb{R}^4$.
However, for a complex line, for example, $3-2z-w=0$, how should we visualize it? Is it still a plane in $\mathbb{R}^4$?
I found that complex hypersurfaces are very difficult to visualize. Even for $\mathbb{C}^2$, the real dimension goes to $4$, beyond the scope of an Earth creature.
Sure it is: $3-2z-w = -(w+1) - 2(z-2)$, so that $\{3-2z-w =0\}$ is the "real plane" $\{-w-2z = 0\}$ shifted by the vector $(-1,2) \in \mathbb C^2$.
Yeah, that's a bit difficult, here is me rambling around:
When dealing with complex (often algebraic) curves, I often visualize the real points. Or think of it a bit like a real curve in $\mathbb R^2$. One advantage is that the intersection of two curves is always a discrete set. Which is not the case if you think of real $2-$folds in $\mathbb R^4$ (or even two-dimensional subvector spaces).
When dealing with compact complex curves ("compact Riemann surfaces"), the visualization as a "torus with many holes" also comes in handy, as it catches the topology. But even then, this visualization doesn't catch everything about the curve. For example elliptic curves are all topologically tori, but not necessarily isomorphic as complex manifolds (biholomorphic).