a conformal map
Picture a fine sheet of graph paper laid over a flat region — the little squares meet at right angles, and a tiny circle drawn anywhere stays a circle. A conformal map is a way of bending and reshaping that region onto another one so that, even though distances and overall shapes get distorted, every crossing angle is preserved exactly. Two roads that meet at 30 degrees still meet at 30 degrees after the map; a tiny square is still (very nearly) a square, just rotated and resized. The grid may curve and stretch, but locally it never gets sheared into a skewed parallelogram.
Precisely, a map f from one open region to another is conformal if it is holomorphic (complex-differentiable everywhere on the region) and its derivative f'(z) is never zero there. The reason angles survive is the secret life of the complex derivative: near a point z_0, the map behaves like z -> f(z_0) + f'(z_0)(z - z_0), and multiplying by the fixed number f'(z_0) is just a rotation (by the angle arg f'(z_0)) combined with a uniform scaling (by the factor |f'(z_0)|). A rotation-and-scaling treats all directions even-handedly, so it cannot change the angle between two curves. This is exactly the 'amplitwist' picture of the derivative.
Conformal maps matter because they let you redraw a hard problem on an easy region without disturbing the laws that live on angles — most importantly Laplace's equation, which governs steady heat, electrostatics, and ideal fluid flow. An honest caution: 'preserves angles' is a strictly local statement, true in the infinitesimal limit at each point. A conformal map can still warp a big square into something wildly curved overall; what it guarantees is only that infinitely small figures keep their shape. And the requirement f'(z) not equal to 0 is essential — where the derivative vanishes, angles are not preserved at all.
The map f(z) = z^2 takes the right-angle corner where the positive x-axis meets the positive y-axis and opens it into a straight angle (180 degrees), because at the corner point z = 0 its derivative f'(z) = 2z is zero. Away from the origin, where f'(z) is not zero, f really is conformal: a tiny grid in the first quadrant maps to a tiny grid whose lines still cross at right angles.
Conformality holds wherever f'(z) is not zero — and visibly breaks at the critical point z = 0.
A common slip is to think 'conformal' means 'shape-preserving' in the everyday sense. It does not: lengths and large-scale shapes change freely. Only angles (and orientation) are kept, and only in the infinitesimal limit.