In the next decade future measurements of the Cosmic Microwave Background anisotropies are expected to reach the sensitivity
to constrain the inverse and direct neutrino hierarchies. A cosmological detection of non-zero neutrino mass seems therefore around
the corner. However one important aspect is often neglected: current predictions are done under the assumption of the standard
$\Lambda$CDM model, while several hints for tensions and anomalies are currently present in current data that stills leave as open
the possibility of new physics.