  In this note we classify all Bonnet pairs on a simply connected domain. Our
main intent was to apply what we call a quaternionic function theory to a
concrete problem in differential geometry. The ideas are simple: conformal
immersions into quaternions or imaginary quaternions take the place of chart
maps for a Riemann surface. Starting from a reference immersion we construct
all conformal immersions of a given (simply connected) Riemann surface (up to
translational periods) by spin transformations. With this viewpoint in mind we
discuss how to construct all Bonnet pairs on a simply connected domain from
isothermic surfaces and vice versa. Isothermic surfaces are solutions to a
certain soliton equation and thus a simple dimension count tells us that we
obtain Bonnet pairs which are not part of any of the classical Bonnet families.
The corresponcence between Bonnet pairs and isothermic surfaces is explicit and
to each isothermic surface we obtain a 4-parameter family of Bonnet pairs.
