  We give a precise and complete description on the spectrum for a class of
non-self-adjoint quasi-periodic operators acting on $\ell^2(\mathbb{Z}^d)$
which contains the Sarnak's model as a special case. As a consequence, one can
see various interesting spectral phenomena including $\mathscr{P}\mathscr{T}$
symmetric breaking, the non-simply-connected two-dimensional spectrum in this
class of operators. Particularly, we provide new examples of non-self-adjoint
operator in $\mathscr{l}^{2}(\mathbb{Z})$ whose spectra (actually a
two-dimensional subset of $\mathbb{C}$) can not be approximated by the spectra
of its finite-interval truncations.
