In the real case, one can also use a Harnack inequality for the linearized equation due to Caffarelli and Gutierrez (here).
The idea goes as follows. If $\det D^2u = 1$, then by the concavity of the equation, $\varphi = \Delta u$ is a positive subsolution to the linearized equation:
$$u^{ij}\varphi_{ij} \geq 0.$$
A version of the mean value inequality holds for subsolutions to this equation, in an ellipsoidal geometry in which the balls are replaced by the level sets of $u$. If $u$ has zero boundary data on a domain in $\mathbb{R}^n$ equivalent to $B_1$ up to dilations, then Caffarelli and Gutierrez show that these level sets have covering properties that are good enough to imitate the proof of the Krylov-Safonov Harnack inequality, giving
$$\varphi(0) \leq C(n) \int_{B_{1/2}} \varphi \,dx$$
(since $B_{1/2}$ contains some reasonably round level set of $u$), and since $\varphi = \Delta u$ and $u$ is locally Lipschitz, one sees that the right side is bounded by integrating by parts.
(Actually, The Harnack inequality for linearized Monge-Ampere only requires that $\det D^2u$ is bounded above and below by positive constants, but we need the right side to be nice to say that $\Delta u$ is a subsolution.)
Both this and the Pogorelov computation are based on on the observation that $\Delta u$ has no local maxima. While this method is more involved, advantages are that it doesn't require one to guess tricky cutoff quantities, and that it gives a helpful geometric understanding of the degeneracy/ affine invariance of the equation. A disadvantage is that it heavily relies on the convexity of $u$, which doesn't hold in e.g. the complex setting.
Also, in two dimensions (real case again) one has special techniques such as the partial Legendre transform (obtained by taking the Legendre transform along lines in some direction $e$). If $\det D^2u = 1$ then its transform $u^*$ is harmonic and $u^*_{ee} > 0$, and one can get an upper bound for $u_{ee}$ by using the Harnack inequality to get a positive lower bound for $u^*_{ee}$.
