We study surfaces in the complexified sphere parameterized by a complex orthogonal net and show that the fundamental theorem of these surfaces is stated in terms of two functions called the curvatures. We also exhibit fundamental examples by assuming natural conditions on the curvatures, which include a portion of the sphere, the hyperboloid of one sheet or the hyperboloid of two sheets, and the surfaces whose integrability condition is given by the complexified Liouville�s equation.