the amplitwist
Here is the most vivid way to picture what f'(z) actually DOES, a coinage from Tristan Needham's geometric approach. Near a point z_0 a holomorphic function acts, to first order, by taking any tiny arrow sticking out of z_0 and doing two things to it: amplifying its length and twisting its direction. The single complex number f'(z_0) packages both operations at once — its modulus |f'(z_0)| is the amplification factor and its argument arg f'(z_0) is the twist (rotation) angle. Needham fuses 'amplify' and 'twist' into the word amplitwist.
Why does multiplying by a complex number do exactly this? Because multiplication by a + i b = r e^(i theta) scales any vector by r = |a + i b| and rotates it by theta = arg(a + i b). Since complex differentiability means f(z) is locally approximated by f(z_0) + f'(z_0)(z - z_0), the displacement z - z_0 gets multiplied by f'(z_0) — and that multiplication is precisely an amplification by |f'(z_0)| together with a rotation by arg f'(z_0). The same amplitwist is applied to EVERY little arrow at z_0, regardless of its initial direction.
This uniformity over all directions is the geometric heart of why holomorphic maps preserve angles (are conformal) wherever f' is nonzero: if every arrow turns by the same angle, the angle BETWEEN two arrows is unchanged. Where f'(z_0) = 0 there is no well-defined twist and conformality fails; such a point is a critical point. The amplitwist picture is also why the Cauchy-Riemann equations geometrically mean 'the local linear map is a rotation-scaling, not a general shear'.
For f(z) = z^2 at z_0 = i: f'(i) = 2i, with modulus 2 and argument pi/2. So near i the map doubles every small length and rotates every small arrow by 90 degrees — a uniform amplitwist applied to all directions at once.
f'(z_0) is one complex number doing two jobs: |f'| amplifies, arg f' twists.
The amplitwist is a first-order (local, infinitesimal) picture; over a finite region a holomorphic map bends and stretches differently from point to point. Where f' = 0, there is no twist and angles are NOT preserved.