Interest in fullerenes has been renewed recently in astrophysics as a consequence of their detection in circumstellar environments. In particular, C-60(+) was detected in the diffuse interstellar medium and its presence has been related to some diffuse interstellar bands (DIBs) whose origin was previously unknown. A single recent laboratory experiment (J. Phys. Chem. A 2017, 121, 7356-7361) shows that upon laser excitation at 785 nm, C-60(+) in neon matrixes exhibits a radiative decay at 965 nm, while UV photoexcitation does not lead to any significant luminescence. To rationalize this original dual photophysical behavior, we have performed time-dependent density functional theory (TD-DFT) calculations on C-60(+) to investigate the potential energy surfaces of the relevant electronic states, completed by the simulations of vibrationally resolved absorption and emission spectra. The proposed photophysical pathways shed light on the experimental measurements: The near-IR laser excitation populates the 11th doublet excited state (D-11) that decays to the lowest first bright excited state D-5, from which photoluminescence is predicted. Indeed, D-5 is largely separated from the lower electronic states (D-0-D-4). Thus, D-5 behaves effectively as the first excited state, while the D-0-D-4 set of states act as the electronic ground state. In addition, there are no low-lying conical intersections between D-5 and lower excited states energetically accessible upon near-IR excitation that can provide efficient nonradiative decay channels for this state, leaving radiative decay as the most likely deactivation pathway. However, a sloped conical intersection between D-5 and D-4 was located around 2.9 eV above D-0. While it is too high in energy to be accessible upon near-IR excitation, it provides a funnel for efficient nonradiative decay down to the ground state (D-0) accessible upon UV light excitation. Thus, the photophysics of C-60(+) is controlled by the ability to access this funnel: Upon near-IR excitation, the system fluoresces because the funnel for nonradiative decay cannot be reached, while UV irradiation provides a different route by opening up a radiationless decay channel via this funnel, accounting for the absence of fluorescence.