This study proposes a multi-dimensional approach to investigate, represent, and categorize students' in-depth understanding of complex physics concepts. Clinical interviews were conducted with 30 undergraduate physics students to probe their understanding of heat conduction. Based on the data analysis, six aspects of the participants' responses were identified as nominal scales and designated as six dimensions in a multi-axis (star) diagram to represent their in-depth understanding of heat conduction. The results demonstrated a wide diversity of the participants' in-depth understanding of heat conduction. In addition, the proportions of participants' naive ideas in the six dimensions were low, and many of them used some viable, sophisticated rules for explaining relevant phenomena of heat conduction. Furthermore, the patterns of the multi-dimensional diagram illustrated that the participants who, across all dimensions, possessed scientifically accepted understanding performed better in the probes of their scientific explanations. This study also discusses the educational and instructional values of this multi-dimensional analysis, and particularly highlights the importance of investigating students' multi-dimensional understanding to more fully account for the large variance in individual differences likely to be encountered in instructional settings. (Contains 2 tables and 2 figures.)