Let h : C ! C be an R-linear map. In this thesis, we explore the dynamics of the quasiregular mapping h(z)2 + c. It is well-known that a polynomial can be conjugated by a holomorphic map to w 7! wd in a neighbourhood of infinity. This map is called a Böttcher coordinate for f near infinity. We construct a Böttcher type coordinate for compositions of h and polynomials, a class of mappings first studied in [19]. As an application, we prove that if h is affne and c 2 C, then h(z)2 + c is not uniformly quasiregular. Via the Böttcher type coordinate, we are able to obtain results for any degree two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur, and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. Finally, we show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.