The Be star zeta Tau was recently reported to be a gamma Cas analog; that is, it displays an atypical (bright and hard) X-ray emission. The origin of these X-rays remains debated.The first X-ray observations indicated a very large absorption of the hot plasma component (N_H~ 10^{23}/cm^2). This is most probably related to the edge-on configuration of the zeta Tau disk. If the X-ray emission arises close to the companion, an orbital modulation of the absorption could be detected as the disk comes in and out of the line of sight. New XMM-Newton data were obtained to characterize the high-energy properties of zeta Tau in more detail. They are complemented by previous Chandra and SRG/eROSITA observations as well as by optical spectroscopy and TESS photometry. The high-quality XMM-Newton data reveal the presence of a faint soft X-ray emission, which appears in line with that recorded for non-gamma Cas Be stars. In addition, zeta Tau exhibits significant short-term variability at all energies, with larger amplitudes at lower frequencies (``red noise''), as is found in X-ray data of other gamma Cas stars. Transient variability (softness dip, low-frequency signal) may also be detected at some epochs. In addition, between X-ray exposures, large variations in the spectra are detected in the 1.5-4.keV energy band. They are due to large changes in absorption toward the hottest (9keV) plasma. These changes are not correlated with either the orbital phase or the depth of the shell absorption of the Halpha line. These observed properties are examined in the light of proposed gamma Cas models.