The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by q-normal operators including generators and the index pairing. The C*-algebras generated by q-normal operators can be viewed as a q-deformation of the quantum complex plane. In this sense, we find deformations of the classical Bott projections describing complex line bundles over the 2-sphere, but there are also simpler generators for the K_0-groups, for instance 1-dimensional Powers-Rieffel type projections and elementary projections belonging to the C*-algebra. The index pairing between these projections and generators of the even K-homology group is computed, and the result is used to express the K_0-classes of the quantized line bundles of any winding number in terms of the other projections.