For example, starting with y = x^2+1:
You can see that the real component has a hyperbolic parabaloid shape. The imaginary component does too, just rotated by 45 degrees. This shape clearly has zeroes, since it's continuous and has infinite range. You can try to look for spots where both components are zero, or you can plot the magnitude:
You can then clearly see that the zeros are "out" and below the global minimum of the real plot.
You can only really look at a slice at a time, but I find this easier than the crumpled-paper-at-every-point approach, because it's a lot harder to see patterns emerge in nearby bits of crumpled paper.
Representing the result of a function from R to R requires two orthogonal axes because the element pre-transformation is 1-dimensional and the result is also 1-dimensional.
For C to R, this would require 2+1=3 orthogonal axes, so it can be visualized with a 3D representation. Likewise for R to C.
From C to C that would be 4-dimensional and becomes already trickier without some effort to conceptualize it, and certainly becomes less intuitive without resorting to alternate ways to conceptualize dimensions.
It quite probable that beyond that, one would really encounter decreasing returns on trying to visualize the situation because the cost of abstraction would increase in order to rely on multisensorial approaches to compensate for our inability to visually perceive much beyond 3D.
It's entirely possible that the whole post flew way over my head and I absolutely did not get it though, in which case I am truly just a confused commenter.
The idea of a mathematical function of a function doesn't sound like a big deal; it sounds vaguely similar to mundane abstractions in programming. This sort of thing is beyond my ability to cope with and yet it sounds like counting 1, 2, 3, not like abracadabra...
I think your point with functional programming is spot on. Just as one learns, to pick that bête noire, monads not by reading yet another clever re-packaging that somehow only manages to make them sounds more difficult, but rather by finding a problem for which they're relevant and getting a feel for them by using them, so too does one learn about infinite-dimensional space not by treating it as some sort of philosophical profundity, but by finding a problem for which it's the right setting, and realising that it's just a mathematical tool like any other.
This perspective could be interpreted as treating equations as recipes and number systems, (fields/rings) as ingredients. It turns out to be extremely fruitful to study the "R-points" of an equation for all sorts of rings R, even more exotic ones like p-adic numbers or finite fields.
Algebraic geometry and the study of schemes is a truly beautiful field of mathematics, linking number theory to theoretical physics, and much more.
A function in the complex plane is a 4-dimensional thing. A graph of that function on the real plane is a 2-D slice of that 4-D thing.
That said, the one visualization of complex numbers that I wish more understood was a complex number in polar coordinates. In polar coordinates, addition is complicated. But multiplication is simple. Every complex number is a magnitude and an angle. You multiply the magnitudes and add the angles.
What this means is that -1 is (1, 180 degrees). Literally a turn halfway around the circle. And now what are its square roots? Well i is (1, 90 degrees) and -i is (1, -90 degrees). Now stand up and actually do those turns.
The result is that i is a turning motion that takes you off the real line. But that visualization helps build intuition about why in the complex plane there should be a close connection between exponential functions and sin/cos. (Specifically e^(ix) = cos(x) + i sin(x) - in other words it is a turn by x radians.)
I think the initial confusion stemmed from not pinning down the domain of x and y.
When he says “I don’t see the complex points” in the xy graph that is because they aren’t there: the xy graph is in R^2 and x^2 + 1 = y has no solution in the reals for y = 0.
But if you imagine, as he did, an extra dimension attached at each point then that is now a complex space, is it not?