Illustration of the complex linear function f(z) = a + b*i, where a = sqrt(3) + i og b = 2 + i.
A mesh of unit squares in the z-plane are mapped by f onto another square net in the w-planet. 
In the z-plane the three complex numbers 0, 1 and i as well as the fixed point z* of the mapping are marked by colored dots and a black cross, respectively.

Their images in the w-plane have the same signatures. The fixed point is defined by the requirement f(z*) = z*, so as the only number it retains its position.
Own work
AstroOgier