The citation impact of a scientific publication is usually seen as a one-dimensional concept. We introduce a multidimensional framework for characterizing the citation impact of a publication. In addition to the level of citation impact, quantified by the number of citations received by a publication, we also conceptualize and operationalize the depth and breadth and the dependence and independence of the citation impact of a publication. The proposed framework distinguishes between publications that have a deep citation impact, typically in a relatively narrow research area, and publications that have a broad citation impact, probably covering a wider area of research. It also makes a distinction between publications that are strongly dependent on earlier work and publications that make a more independent scientific contribution. We use our multidimensional citation impact framework to report basic descriptive statistics on the citation impact of highly cited publications in all scientific disciplines. In addition, we present a detailed case study focusing on the field of scientometrics. The proposed citation impact framework provides a more in-depth understanding of the citation impact of a publication than a traditional one-dimensional perspective.