I find this less intuitive than simply looking at the real component, imaginary component, magnitude, and phase plots of simple, and then progressively more com

I find this less intuitive than simply looking at the real component, imaginary component, magnitude, and phase plots of simple, and then progressively more complicated functions of interest. For example, starting with y = x^2+1: https://www.wolframalpha.com/input/?i=plot+z%5E2+%2B+1 https://www.wolframalpha.com/input/?i=plot+z%5E2+%2B+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: https://www.wolframalpha.com/input/?i=plot+Magnitude%28z%5E2+%2B+1%29+from+-2i+to+2i https://www.wolframalpha.com/input/?i=plot+Magnitude%28z%5E2... 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.