We present gravitational lens models of the multiply imaged quasar DES J0408-5354, recently discovered in the Dark Energy Survey (DES) footprint, with the aim of interpreting its remarkable quad-like configuration. We first model the DES single-epoch $grizY$ images as a superposition of a lens galaxy and four point-like objects, obtaining spectral energy distributions (SEDs) and relative positions for the objects. Three of the point sources (A,B,D) have SEDs compatible with the discovery quasar spectra, while the faintest point-like image (G2/C) shows significant reddening and a `grey dimming of $approx0.8$mag. In order to understand the lens configuration, we fit different models to the relative positions of A,B,D. Models with just a single deflector predict a fourth image at the location of G2/C but considerably brighter and bluer. The addition of a small satellite galaxy ($R_{rm E}approx0.2$) in the lens plane near the position of G2/C suppresses the flux of the fourth image and can explain both the reddening and grey dimming. All models predict a main deflector with Einstein radius between $1.7$ and $2.0,$ velocity dispersion $267-280$km/s and enclosed mass $approx 6times10^{11}M_{odot},$ even though higher resolution imaging data are needed to break residual degeneracies in model parameters. The longest time-delay (B-A) is estimated as $approx 85$ (resp. $approx125$) days by models with (resp. without) a perturber near G2/C. The configuration and predicted time-delays of J0408-5354 make it an excellent target for follow-up aimed at understanding the source quasar host galaxy and substructure in the lens, and measuring cosmological parameters. We also discuss some lessons learnt from J0408-5354 on lensed quasar finding strategies, due to its chromaticity and morphology.
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