  We present the analysis of Chandra and NuSTAR spectra of NGC 4968, a local
(D$\sim$44 Mpc) 12$\mu$m-selected Seyfert 2 galaxy, enshrouded within
Compton-thick layers of obscuring gas. We find no evidence of variability
between the Chandra and NuSTAR observations (separated by 2 years), and between
the two NuSTAR observations (separated by 10 months). Using self-consistent
X-ray models, we rule out the scenario where the obscuring medium is nearly
spherical and uniform, contradicting the results implied by the $<$10 keV
Chandra spectrum. The line-of-sight column density, from intervening matter
between the source and observer that intercepts the intrinsic AGN X-ray
emission, is well within the Compton-thick regime, with a minimum column
density of $2\times10^{24}$ cm$^{-2}$. The average global column density is
high ($> 3\times10^{23}$ cm$^{-2}$), with both Compton-thick and Compton-thin
solutions permitted depending on the X-ray spectral model. The spectral models
provide a range of intrinsic AGN continuum parameters and implied 2-10 keV
luminosities ($L_{\rm 2-10keV,intrinsic}$), where the higher end of $L_{\rm
2-10keV,intrinsic}$ is consistent with expectations from the 12$\mu$m
luminosity ($L_{\rm 2-10keV,intrinisc} \sim 7\times10^{42}$ erg s$^{-1}$).
Compared with Compton-thick AGN previously observed by {\it NuSTAR}, NGC 4968
is among the most intrinsically X-ray luminous. However, despite its close
proximity and relatively high intrinsic X-ray luminosity, it is undetected by
the 105 month Swift-BAT survey, underscoring the importance of multi-wavelength
selection for obtaining the most complete census of the most hidden black
holes.
