This chapter begins with a few remarks indicating briefly the motivation for studying quantum theory from a geometrical point of view, and in particular for the development of a non-linear geometrical quantum theory. It considers the case of an elementary quantum mechanical system consisting of a spin one-half particle in a constant magnetic field. The successes of quantum mechanics have been so wide and varied, that one must be wary of making any gross alteration to the structure of the theory. The connection with twistor theory goes perhaps one level deeper, since the particular geometrical characteristics of quantum theory that emerge as being relevant have to do precisely with the complex analytic structure of the quantum mechanical state manifold. A Kahler manifold can either be regarded as a real Riemannian manifold that is endowed also with a complex structure, or as a complex manifold that is also endowed with a Riemannian structure.