  This study presents the capabilities of the AXIS telescope in estimating
redshifts from X-ray spectra alone (X-ray redshifts, XZs). Through extensive
simulations, we establish that AXIS observations enable reliable XZ estimates
for more than 5500 obscured Active Galactic Nuclei (AGN) up to redshift $z\sim
6$ in the proposed deep (7 Ms) and intermediate (375 ks) surveys. Notably, at
least 1600 of them are expected to be in the Compton-Thick regime ($\log
N_H/\mathrm{cm^{-2}}\geq 24$), underscoring the pivotal role of AXIS in sample
these elusive objects that continue to be poorly understood. XZs provide an
efficient alternative for optical/infrared faint sources, overcoming the need
for time-consuming spectroscopy, potential limitations of photometric
redshifts, and potential issues related to multi-band counterpart association.
This approach will significantly enhance the accuracy of constraints on the
X-ray luminosity function and obscured AGN fractions up to high redshift. This
White Paper is part of a series commissioned for the AXIS Probe Concept
Mission; additional AXIS White Papers can be found at the AXIS website
(http://axis.astro.umd.edu) with a mission overview here: arXiv:2311.00780.
