This paper introduces a measure of communication complexity for a two-agent distributed control system where controls are subject to finite bandwidth communication constraints. The proposed complexity measure is an extension of the idea of communication complexity defined in distributed computing. Applying this classical concept to control problems with finite communication constraints leads to a new perspective and a host of new questions, some of which are investigated in this paper. In particular, one can connect the proposed complexity with the traditional communication complexity via upper bound and lower bound inequalities. Moreover, the proposed complexity is shown to be intricately related to the dynamical characteristics of the underlying system.