Bézier curves and the Takagi function

We consider B\'ezier curves with complex parameter, and we determine explicitly the affine iterated function system (IFS) corresponding to the de Casteljau subdivision algorithm, together with the complex parametric domain over which such an IFS has a unique global connected attractor. For a specific family of complex parameter having vanishing imaginary part, we prove that the Takagi fractal curve is the attractor, under suitable scaling.