We re-analyze 4 years of HARPS spectra of the nearby M1.5 dwarf GJ 667C available through the European Southern Observatory public archive. The new radial velocity (RV) measurements were obtained using a new data analysis technique that derives the Doppler measurement and other instrumental effects using a least-squares approach. Combining these new 143 measurements with 41 additional RVs from the Magellan/Planet Finder Spectrograph and Keck/High Resolution Echelle Spectrometer spectrometers reveals three additional signals beyond the previously reported 7.2 day candidate, with periods of 28 days, 75 days, and a secular trend consistent with the presence of a gas giant (period similar to 10 years). The 28 day signal implies a planet candidate with a minimum mass of 4.5M. orbiting well within the canonical definition of the star's liquid water habitable zone (HZ), that is, the region around the star at which an Earth-like planet could sustain liquid water on its surface. Still, the ultimate water supporting capability of this candidate depends on properties that are unknown such as its albedo, atmospheric composition, and interior dynamics. The 75 day signal is less certain, being significantly affected by aliasing interactions among a potential 91 day signal, and the likely rotation period of the star at 105 days detected in two activity indices. GJ 667C is the common proper motion companion to the GJ 667AB binary, which is metal-poor compared to the Sun. The presence of a super-Earth in the HZ of a metal-poor M dwarf in a triple star system supports the evidence that such worlds should be ubiquitous in the Galaxy.