We present ground-based broad-band photometry of two transits in the WASP-44 planetary system obtained simultaneously through four optical (Sloan g', r', i', z') and three near-infrared (NIR; J, H, K) filters. We achieved low scatters of 1-2mmag per observation in the optical bands with a cadence of roughly 48s, but the NIR-band light curves present much greater scatter. We also observed another transit of WASP-44 b by using a Gunn r filter and telescope defocussing, with a scatter of 0.37 mmag per point and an observing cadence around 135 s. We used these data to improve measurements of the time of mid- transit and the physical properties of the system. In particular, we improved the radius measurements of the star and planet by factors of 3 and 4, respectively. We find that the radius of WASP-44 b is 1.002+/-0.033+/-0.018RJup (statistical and systematic errors, respectively), which is slightly smaller than previously thought and differs from that expected for a core-free planet. In addition, with the help of a synthetic spectrum, we investigated the theoretically predicted variation of the planetary radius as a function of wavelength, covering the range 370-2440nm. We can rule out extreme variations at optical wavelengths, but unfortunately our data are not precise enough (especially in the NIR bands) to differentiate between the theoretical spectrum and a radius which does not change with wavelength. The resulting measurements of transit mid-points were fitted with a straight line to obtain a new orbital ephemeris: T0=BJD(TDB)2455434.37642(37)+2.4238133(23)xE, where E is the number of orbital cycles after the reference epoch [the mid-point of the first transit observed by Anderson et al. (2012, Cat. J/MNRAS/422/1988)] and quantities in parentheses denote the uncertainty in the final digit of the preceding number.