  We use Krylov complexity to study operator growth in the $q$-body dissipative
SYK model, where the dissipation is modeled by linear and random $p$-body
Lindblad operators. In the large $q$ limit, we analytically establish the
linear growth of two sets of coefficients for any generic jump operators. We
numerically verify this by implementing the bi-Lanczos algorithm, which
transforms the Lindbladian into a pure tridiagonal form. We find that the
Krylov complexity saturates inversely with the dissipation strength, while the
dissipative timescale grows logarithmically. This is akin to the behavior of
other $\mathfrak{q}$-complexity measures, namely out-of-time-order correlator
(OTOC) and operator size, which we also demonstrate. We connect these
observations to continuous quantum measurement processes. We further
investigate the pole structure of a generic auto-correlation and the
high-frequency behavior of the spectral function in the presence of
dissipation, thereby revealing a general principle for operator growth in
dissipative quantum chaotic systems.
