  In this paper, we establish a local regularity result for
$W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic
equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the
Monge-Amp\`ere equation. Notably, we apply our result to the so-called
$k$-Monge-Amp\`ere equation.
