  We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two
$\hat{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings,
one the Hermitian conjugate of the other, living in off-diagonal blocks of a
larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero
energy. We employ diagrammatic and combinatorial techniques to compute
analytically the low-order moments of the Hamiltonian. In the limit of large
number $N$ of Majoranas, we have found striking similarities with the moments
of the weight function of the Al-Salam-Chihara $Q$-Laguerre polynomials. For
$\hat{q} = 3, 4$, the $Q$-Laguerre prediction, with $Q=Q(\hat{q},N)$ also
computed analytically, agrees well with exact diagonalization results for $30 <
N \leq 34$ while we observe some deviations for $\hat q = 2$. The most salient
feature of the spectral density is that, for odd $\hat{q}$, low-energy
excitations grow as a stretched exponential, with a functional form different
from that of the supersymmetric SYK model. For $\hat q = 4$, a detailed
analysis of level statistics reveals quantum chaotic dynamics even for time
scales substantially shorter than the Heisenberg time. More specifically, the
spacing ratios in the bulk of the spectrum and the microscopic spectral density
and the number variance close to the hard edge are very well approximated by
that of an ensemble of random matrices that, depending on $N$, belong to the
chiral or superconducting universality classes. In particular, we report the
first realization of level statistics belonging to the chGUE universality
class, which completes the tenfold-way classification in the SYK model.
