Aims. We present new data for four candidate obscured Compton-Thick (CT) quasars at z ∼ 1-2.5 observed with the SINFONI VLT spectrograph in adaptive optics (AO) mode. These sources were selected from a 24 µm Spitzer MIPS survey of the COSMOS field, on the basis of red mid-infrared to optical and optical to near-infrared colours, with the intention of identifying active galactic nuclei (AGNs) in dust enshrouded environments, where most of the black hole mass is assembled. Methods. Near-infrared spectra were analysed to check for emission line features and to search for broad components in the [OIII]-Hβ and Hα-[NII] regions. We also employed X-ray spectral analysis, radio and MIR diagnostics, and SED fitting to study the nature of the sources. Results. We successfully identified three objects for which we had only a photometric redshift estimate. Based on their emission line diagnostics and on ancillary multi-wavelength constraints, we find that all four targets harbour obscured AGNs. Broad profiles, which could be attributed to the effects of outflows, are revealed in only one target, MIRO20581. In particular, we clearly resolved a fast (∼1600 km s −1 ) and extended (∼5 kpc) outflow in the [OIII]5007 emission line. This feature, the commonly used indicator for ionised outflowing gas, was only sampled and detected for this target; hence, we cannot exclude the presence of outflows in the other sources. Overall, the constraints we obtain from our targets and from other comparative samples from the literature suggest that these optically faint luminous infrared galaxies, hosting obscured AGNs, may represent a brief evolutionary phase between the post-merger starburst and the unobscured quasar phases.