Magnetic fields play a crucial role for the physics of the dilute and hot upper parts of the solar atmosphere, from the low chromosphere up to the outer corona. Numerical modeling of these regions by MHD codes and 3D radiative transfer in spectral lines has started, we urgently need direct measurement of the magnetic and velocity fields in order to confirm and improve such modeling. Direct measurements of the magnetic field vector rely on spectro-polarimetric observations of the Hanle or Zeeman effects in spectral lines. In the upper solar atmosphere the magnetic field strength tends to decrease with height and lines are broaedened both thermally and by turbulent and wave motions. The sensitivity of spectral lines to Zeeman effect is thus low under such conditions, but it increases linearly with wavelength. The infrared is the domain of choice for Zeeman observations in chromospheric and coronal lines. Moreover, as chromospheric and coronal structures develop on a broad range of spatial scales, from very fine to large scales, high spatial resolution observations are required. These observations are very demanding in photometric sensitivity, together with spatial and temporal resolution. Dome C conditions of both very good day-time seeing and high coronal quality are unique and make it a site of choice for a possible breakthrough in our understanding of the physical mechanisms at play in the outer solar atmosphere.