the rotation-and-scaling map a z
Multiplying by a fixed complex number, f(z) = a z, does two simple things at once: it rotates the plane and it scales it. This is the geometric meaning of complex multiplication itself. If you write the constant in polar form, a = r e^(i alpha), then multiplying z by a multiplies its modulus by r (a stretch or shrink) and adds alpha to its argument (a rotation about the origin by angle alpha).
To see it, take any z and write it as z = R e^(i theta). Then a z = (r e^(i alpha))(R e^(i theta)) = (r R) e^(i (theta + alpha)). The new length is r R and the new angle is theta + alpha — the length got multiplied by r, the angle got increased by alpha. A pure rotation is the case r = 1 (a = e^(i alpha)); a pure scaling is the case alpha = 0 (a is a positive real number). Multiplying by i, for instance, is a 90-degree rotation, because i = e^(i pi/2).
Because it preserves angles between curves and multiplies all lengths by the same factor, a z is the prototype of a similarity transformation, and the simplest non-trivial conformal map. It keeps the origin fixed (a times 0 is 0). Combined with translations it gives all the maps f(z) = a z + b, the linear maps, which already capture every rigid motion and similarity of the plane and form the backbone of the Mobius transformations.
Take a = 1 + i = sqrt(2) e^(i pi/4). Then f(z) = (1 + i) z rotates the whole plane 45 degrees counter-clockwise and stretches every length by sqrt(2). The point 1 maps to 1 + i; the point i maps to -1 + i.
The modulus of a scales lengths; the argument of a rotates the plane — two effects in one multiplication.
The scaling factor and rotation angle are the same everywhere — that uniformity is exactly why a z is conformal. By contrast z^2 also rotates and scales, but by an amount that varies from point to point, so it is conformal only away from the origin.