We present detailed modelling of the recently discovered, quadruply lensed quasar J0408-5354, with the aim of interpreting its remarkable configuration: besides three quasar images (A, B, D) around the main deflector (G1), a fourth image (C) is significantly reddened and dimmed by a perturber (G2) which is not detected in the Dark Energy Survey imaging data. From lens models incorporating (dust-corrected) flux ratios, we find a perturber Einstein radius 0.04 arcsec less than or similar to R-E,R- G2 less than or similar to 0.2 arcsec and enclosed mass M-p(R-E,R- G2) less than or similar to 1.0 x 10(10) M-circle dot. The main deflector has stellar mass log(10)(M-star/M-circle dot) = 11.49(-0.32)(+0.46), a projected mass M-p(R-E,R- G1) approximate to 6 x 10(11) M-circle dot within its Einstein radius R-E,R- G1 = (1.85 +/- 0.15) arcsec and predicted velocity dispersion 267-280 km s(-1). Follow-up images from a companion monitoring campaign show additional components, including a candidate second source at a redshift between the quasar and G1. Models with free perturbers, and dust-corrected and delay-corrected flux ratios, are also explored. The predicted time-delays (Delta t(AB) = (135.0 +/- 12.6) d, Delta t(BD) = (21.0 +/- 3.5) d) roughly agree with those measured, but better imaging is required for proper modelling and comparison. We also discuss some lessons learnt from J0408-5354 on lensed quasar finding strategies, due to its chromaticity and morphology.