Aims: We present a detailed near-infrared/optical/UV study of the transient low-mass X-ray binary 1RXS J180408.9-342058 performed during its 2015 outburst, which is aimed at determining the nature of its companion star. Methods: We obtained three optical spectra (R ~ 1000) at the 2.1 m San Pedro Mártir Observatory telescope (México). We performed optical and NIR photometric observations with both the REM telescope and the New Technology Telescope (NTT) in La Silla. We obtained optical and UV observations from the Swift archive. Finally, we performed optical polarimetry of the source using the EFOSC2 instrument mounted on the NTT. Results: The optical spectrum of the source is almost featureless since the hydrogen and He I emissions lines, typically observed in LMXBs, are not detected. Similarly, carbon and oxygen lines are not observed either. We marginally detect the He II 4686 Å emission line, suggesting the presence of helium in the accretion disc. No significant optical polarisation level was observed. Conclusions: The lack of hydrogen and He I emission lines in the spectrum implies that the companion is likely not a main-sequence star. Driven by the tentative detection of the He II 4686 Å emission line, we suggest that the system could harbour a helium white dwarf. If this is the case, 1RXS J180408.9-342058 would be an ultra-compact X-ray binary. By combining an estimate of the mass accretion rate together with evolutionary tracks for a He white dwarf, we obtain a tentative orbital period of ~40 min. We also built the NIR-optical-UV spectral energy distribution (SED) of the source at two different epochs. One SED was gathered when the source was in the soft X-ray state and this SED is consistent with the presence of a single thermal component. The second SED, obtained when the source was in the hard X-ray state, shows a thermal component along with a tail in the NIR, which likely indicates the presence of a (transient) jet. Based on observations made with ESO Telescopes at the La Silla Observatory under programme ID 094.D-0692(B).The spectrum shown in Fig. 1 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/587/A102