Extremely luminous, ‘super-Chandrasekhar’ (SC) Type Ia Supernovae (SNe Ia) are as yet an unexplained phenomenon. We analyse a well-observed SN of this class, SN 2009dc, by modelling its photospheric spectra with a spectral synthesis code, using the technique of ‘Abundance Tomography’. We present spectral models based on different density profiles, corresponding to different explosion scenarios, and discuss their consistency. First, we use a density structure of a simulated explosion of a 2 M⊙ rotating C–O white dwarf, which is often proposed as a possibility to explain SC SNe Ia. Then, we test a density profile empirically inferred from the evolution of line velocities (blueshifts). This model may be interpreted as a core-collapse SN with an ejecta mass of ∼3 M⊙. Finally, we calculate spectra assuming an ‘interaction scenario’. In such a scenario, SN 2009dc would be a standard white dwarf (WD) explosion with a normal intrinsic luminosity, and this luminosity would be augmented by interaction of the ejecta with a H-/He-poor circumstellar medium. We find that none of the models tested easily explains SN 2009dc. With the 2 M⊙ WD model, our abundance analysis predicts small amounts of burning products in the intermediate-/high-velocity part of the ejecta (v ≳ 9000 km s−1). However, in the original explosion simulations, where the nuclear energy release per unit mass is large, burned material is present at high velocities. This contradiction can only be resolved if asymmetries strongly affect the radiative transfer or if C–O white dwarfs with masses significantly above 2 M⊙ exist. In a core-collapse scenario, low velocities of Fe-group elements are expected, but the abundance stratification in SN 2009dc seems ‘SN-Ia-like’. The interaction-based model looks promising, and we have some speculations on possible progenitor configurations. However, radiation-hydrodynamics simulations will be needed to judge whether this scenario is realistic at all.