The recent detection of a double red giant branch (RGB) in the optical color-magnitude diagram of the bulge globular cluster HP 1-a more populated, metal-poor, steep giant branch corresponding to the cluster itself and another that is metal rich and curved-led us to explore in the present Letter the possibility that field stars are captured by a globular cluster orbiting in dense bulge regions over several gigayears. Analytical arguments, as well as N-body calculations for a cluster model of 10(5) M. in a bulgelike environment, suggest that a significant fraction of cluster stars may consist of captures. Metal-poor globular clusters in the inner bulge, like HP 1, contrasting in Delta[Fe/H] by at least 1.0 dex with respect to the surrounding metal-rich stars, are ideal probes to further test the capture scenario. In turn, if this scenario is confirmed, the double RGB of HP 1 could provide direct estimates of blanketing amounts, which is fundamental for the photometric calibration of metal-rich stellar populations.