We present a study of the environment of 27 z=3-4.5 bright quasars from the MUSE Analysis of Gas around Galaxies (MAGG) survey. With medium-depth MUSE observations (4 hours on target per field), we characterise the effects of quasars on their surroundings by studying simultaneously the properties of extended gas nebulae and Lyalpha emitters (LAEs) in the quasar host haloes. We detect extended (up to ~ 100 kpc) Lyalpha emission around all MAGG quasars, finding a very weak redshift evolution between z=3 and z=6. By stacking the MUSE datacubes, we confidently detect extended emission of CIV and only marginally detect extended HeII up to ~40 kpc, implying that the gas is metal enriched. Moreover, our observations show a significant overdensity of LAEs within 300 km/s from the quasar systemic redshifts estimated from the nebular emission. The luminosity functions and equivalent width distributions of these LAEs show similar shapes with respect to LAEs away from quasars suggesting that the Lyalpha emission of the majority of these sources is not significantly boosted by the quasar radiation or other processes related to the quasar environment. Within this framework, the observed LAE overdensities and our kinematic measurements imply that bright quasars at z=3-4.5 are hosted by haloes in the mass range ~ 10^{12.0}-10^{12.5} Msun.
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