The processing of signals on simplicial and cellular complexes defined by nodes, edges, and higher-order cells has recently emerged as a principled extension of graph signal processing for signals supported on more general topological spaces. However, most works so far have considered signal processing problems for signals associated to only a single type of cell such as the processing of node signals, or edge signals, by considering an appropriately defined shift operator, like the graph Laplacian or the Hodge Laplacian. Here we introduce the Dirac operator as a novel kind of shift operator for signal processing on complexes. We discuss how the Dirac operator has close relations but is distinct from the Hodge-Laplacian and examine its spectral properties. Importantly, the Dirac operator couples signals defined on cells of neighboring dimensions in a principled fashion. We demonstrate how this enables us, e.g., to leverage node signals for the processing of edge flows.