Context. Research on supernovae (SNe) over the past decade has confirmed that there is a distinct class of events which are much more luminous (by 2 mag) than canonical core-collapse SNe (CCSNe). These events with visual peak magnitudes ≲-21 are called superluminous SNe (SLSNe). The mechanism that powers the light curves of SLSNe is still not well understood. The proposed scenarios are circumstellar interaction, the emergence of a magnetar after core collapse, or disruption of a massive star through pair production. Aims: There are a few intermediate events which have luminosities between these two classes. They are important for constraining the nature of the progenitors of these two different populations and their environments and powering mechanisms. Here we study one such object, SN 2012aa. Methods: We observed and analysed the evolution of the luminous Type Ic SN 2012aa. The event was discovered by the Lick Observatory Supernova Search in an anonymous galaxy (z ≈ 0.08). The optical photometric and spectroscopic follow-up observations were conducted over a time span of about 120 days. Results: With an absolute V-band peak of - 20 mag, the SN is an intermediate-luminosity transient between regular SNe Ibc and SLSNe. SN 2012aa also exhibits an unusual secondary bump after the maximum in its light curve. For SN 2012aa, we interpret this as a manifestation of SN-shock interaction with the circumstellar medium (CSM). If we assume a 56Ni-powered ejecta, the quasi-bolometric light curve requires roughly 1.3 M⊙ of 56Ni and an ejected mass of 14M⊙. This also implies a high kinetic energy of the explosion, 5.4 × 1051 erg. On the other hand, the unusually broad light curve along with the secondary peak indicate the possibility of interaction with CSM. The third alternative is the presence of a central engine releasing spin energy that eventually powers the light curve over a long time. The host of SN 2012aa is a star-forming Sa/Sb/Sbc galaxy. Conclusions: Although the spectral properties of SN 2012aa and its velocity evolution are comparable to those of normal SNe Ibc, its broad light curve along with a large peak luminosity distinguish it from canonical CCSNe, suggesting that the event is an intermediate-luminosity transient between CCSNe and SLSNe at least in terms of peak luminosity. In comparison to other SNe, we argue that SN 2012aa belongs to a subclass where CSM interaction plays a significant role in powering the SN, at least during the initial stages of evolution.