The citation impact of a scientific publication is usually seen as a one-dimensional concept. We introduce a multi-dimensional framework on 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. This enables us to distinguish between publications that have a deep impact concentrated in one specific research field and publications that have a broad impact extending beyond a single research field. It also allows us to make a distinction between publications that are strongly dependent on earlier work and publications that make a more independent scientific contribution. Based on our multi-dimensional citation impact framework, we analyze the citation impact of highly cited publications in all scientific disciplines. In addition, we present a case study focusing on the field of scientometrics. The proposed citation impact framework provides a more detailed understanding of the citation impact of a publication than a traditional one-dimensional perspective.