The short GRB 120323A had the highest flux ever detected with the Gamma-Ray Burst Monitor on board the Fermi Gamma-Ray Space Telescope. Here we study its remarkable spectral properties and their evolution using two spectral models: (1) a single emission component scenario, where the spectrum is modeled by the empirical Band function (a broken power law), and (2) a two-component scenario, where thermal (a Planck-like function) emission is observed simultaneously with a non-thermal component (a Band function). We find that the latter model fits the integrated burst spectrum significantly better than the former, and that their respective spectral parameters are dramatically different: when fit with a Band function only, the E _peak of the event is unusually soft for a short gamma-ray burst (GRB; 70 keV compared to an average of 300 keV), while adding a thermal component leads to more typical short GRB values (E _peak ~ 300 keV). Our time-resolved spectral analysis produces similar results. We argue here that the two-component model is the preferred interpretation for GRB 120323A based on (1) the values and evolution of the Band function parameters of the two component scenario, which are more typical for a short GRB, and (2) the appearance in the data of a significant hardness-intensity correlation, commonly found in GRBs, when we employee two-component model fits; the correlation is non-existent in the Band-only fits. GRB 110721A, a long burst with an intense photospheric emission, exhibits the exact same behavior. We conclude that GRB 120323A has a strong photospheric emission contribution, observed for the first time in a short GRB. Magnetic dissipation models are difficult to reconcile with these results, which instead favor photospheric thermal emission and fast cooling synchrotron radiation from internal shocks. Finally, we derive a possibly universal hardness-luminosity relation in the source frame using a larger set of GRBs ({L_i^{Band}}={(1.59+/- 0.84) \times 10^{50} (E_{peak,i}^{rest})^{1.33+/- 0.07}\ erg\ s^{-1}}), which could be used as a possible redshift estimator for cosmology.