I think about it quite differently. 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 tha

I think about it quite differently. 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.)