In a 2-body system on the complex plane, can't you basically take any point as the origin and describe the motion of te bodies in terms of a complex Fourier ser

In a 2-body system on the complex plane, can't you basically take any point as the origin and describe the motion of te bodies in terms of a complex Fourier series, with the interpretation that the bodies orbit a circle on a circle ... on a circle ... around your chosen point? Edit: Basically Ptolomy did that the 2nd century AD.