the mapping viewpoint
The mapping viewpoint is the decision to picture a complex function not as a graph but as a machine that moves the plane. You set down two copies of the plane, a z-plane and a w-plane, and ask: as z roams over its domain, where does w = f(z) go? A function then becomes a transformation — it picks up each point and sets it down somewhere in the second picture, carrying whole curves and regions with it.
Why insist on this? Because the natural graph would need four dimensions and cannot be drawn, but the action on the plane can. So instead of plotting heights, you draw before-and-after pictures: take a familiar shape in the z-plane (a horizontal line, a circle, a square grid) and draw its image in the w-plane. For f(z) = z^2 a ray from the origin at angle theta lands on a ray at angle 2 theta and its length is squared — so f(z) = z^2 doubles angles at the origin and stretches distances. You learn the function by learning what it does to shapes.
This is the visual soul of complex analysis. Conformal maps (which preserve angles), Mobius transformations (which send circles to circles), the Riemann mapping theorem (which reshapes one region into a disk) — all are statements about how f rearranges the plane. Training yourself to think 'this function is a motion of the plane' makes the whole subject geometric rather than merely algebraic.
Under f(z) = 2z the unit circle |z| = 1 maps to the circle |w| = 2, and a small square doubles in size. Drawing that before-and-after pair tells you instantly that this f is a pure scaling by 2.
Read a complex function by drawing the image of a few familiar shapes, not by plotting a graph.
The mapping picture is intuition, not a replacement for computation. To know exactly where a curve goes you still substitute and simplify; the picture guides you and catches mistakes, but the algebra confirms it.