In reply to lowersharpnose: It's a great idea to look at the absolute value of the function. With z = x + yi, and f(z) = z^2+4 it gives a sort of bowl with a wave around the x-axis. On each line of constant imaginary part of z, the curve is like x^2 + c, where c is a constant, with a minimum on the y-axis (x = 0).
As y reduces towards 2, the bottom of the bowl reduces in minimum height, until it reaches 0 (at y=2), then bounces up again to peak at 4 (y=0), before reducing again to reach 0 at y=-2, then bounces up again and gets further away from the ground again.
Alternatively, think of a conventional bowl, touching the ground at the point (0,0). Do a sort of upward karate chop at this point, leaving the bowl touching the ground at (0,2) and (0,-2) with an upward wave-like curve where it used to touch the ground!