Aims. We seek to constrain the formation of the Galactic bulge by analysing the detailed chemical composition of a large sample of red clump stars in Baade’s window. These stars were selected to minimise the contamination by other Galactic components, so they are good tracers of the bulge metallicity distribution in Baade’s window, at least for stars more metal-rich than ∼−1.5. Methods. We used an automatic procedure to measure [Fe/H] differentially with respect to the metal-rich star μLeo in a sample of 219 bulge red clump stars from R = 20 000 resolution spectra obtained with FLAMES/GIRAFFE at the VLT. For a subsample of 162 stars, we also derived [Mg/H] from spectral synthesis around the Mg i triplet at λ 6319 Å. Results. The Fe and Mg metallicity distributions are both asymmetric with median values of +0.16 and +0.21, respectively. They show only a small proportion of stars at low metallicities, extending down to [Fe/H] = −1.1 or [Mg/H] = −0.7. The iron distribution is clearly bimodal, as revealed both by a deconvolution (from observational errors) and a Gaussian decomposition. The decomposition of the observed Fe and Mg metallicity distributions into Gaussian components yields two populations of equal sizes (50% each): a metal-poor component centred on [Fe/H] = −0.30 and [Mg/H] = −0.06 with a large dispersion and a narrow metal-rich component centred on [Fe/H] = +0.32 and [Mg/H] = +0.35. The metal-poor component shows high [Mg/Fe] ratios (around 0.3), while stars in the metal-rich component are found to have nearly solar ratios. Kinematical differences between the two components have also been found: the metal-poor component shows kinematics compatible with an old spheroid, while the metal-rich component is consistent with a population supporting a bar. In view of their chemical and kinematical properties, we suggest different formation scenarii for the two populations: a rapid formation time scale as an old spheroid for the metal-poor component (old bulge) and for the metal-rich component, a formation on a longer time scale driven by the evolution of the bar (pseudo-bulge). The observations are described well by a simple model consisting of two components: a simple closed box model to predict the metal-poor population contribution and a local thin disc metallicity distribution, shifted in metallicity, to represent the metal-rich population. The pseudo-bulge is compatible with its being formed from the inner thin disc, assuming high (but plausible) values of the gradients in the early Galactic disc.