The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by $q$-normal operators, including generators and 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, one 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 quantized line bundles of any winding number in terms of the other projections.