  It is shown fields that cannot be represented over one complex plane can be
further decomposed for representation over multiple complex planes. This
finding is demonstrated here by solving of the Schr\"{o}dinger equation for the
hydrogen atom in a complex space containing three complex planes. The complex
coordinate system is generated from real coordinates using an isometric
transformation. One plane is applied to mix energy and time; the other two
planes are used to represent the z-component of angular momentum of the
electron. The eigensolutions of the Schr\"{o}dinger equation are shown to be
holomorphic in the complex planes.
