the squaring map z^2
The squaring map f(z) = z^2 is the first complex function that genuinely bends the plane. In polar form its behaviour is transparent: if z = r e^(i theta) then z^2 = r^2 e^(i 2 theta). So squaring doubles the angle every point makes with the positive real axis and squares its distance from the origin. A ray from the origin at angle theta is swung to a ray at angle 2 theta; a circle of radius r becomes a circle of radius r^2.
Doubling the angle has a striking consequence: the map wraps the plane twice around itself. As z sweeps the upper half-plane (angles 0 to pi), the image w = z^2 sweeps angles 0 to 2 pi — a full turn — so the upper half-plane alone already covers the entire w-plane. Run z over the whole plane and every non-zero target is hit twice, because z and -z have the same square. This 'two-to-one' character is why the inverse, the square root, is two-valued and needs a branch cut to tame.
Near the origin the doubling of angles means z^2 is not conformal there: a right angle between two curves through 0 opens up into a straight angle in the image. Everywhere else (z not 0) the map is conformal — locally it looks like a rotation-and-scaling by the factor 2z, the derivative. The origin is its one critical point. Squaring is the simplest window onto branch points, multivaluedness, and how nice maps can still fold the plane.
The quarter-plane x > 0, y > 0 (angles 0 to pi/2) maps under z^2 onto the upper half-plane (angles 0 to pi): the corner at the origin is doubled from a right angle into a straight angle, showing the map is not conformal at 0.
z^2 doubles angles and squares distances, wrapping the plane twice and folding the corner at the origin.
z^2 is conformal everywhere except at z = 0, where its derivative 2z vanishes. A vanishing derivative is precisely the signal of a critical point, where the angle-preserving property breaks down.