Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relation between degree one generalized complex rational B\'ezier curves and quadratic rational B\'ezier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus a rational quadratic planar map can be made to be birational by suitably choosing the middle B\'ezier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to be hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.