What is the definition of a complex manifold with boundary?

Can anybody help me to be clear about this definition. I know the definition of a real manifold with boundary (as in Lee's book) and the definition of a complex manifold (locally diffeomophic to an open set in $\mathbb{C}^{n}$ and transition maps are holomorphic).

What is the definition of a complex manifold with boundary? I see it many times while reading about the complex-Monge Ampere equations on Kahler manifolds.

The usual definition is that $M$ is a complex manifold with boundary if and only if it has an atlas of biholomorphically compatible charts, each of which has as its image either an open subset of $\mathbb C^n$ or a set of the form $\{z\in U :f(z)\le 0\}$, where $U\subseteq\mathbb C^n$ is open and $f\colon U\to\mathbb R$ is a $C^\infty$ submersion.

It's important to allow such "curved" model boundaries instead of insisting that the image be a relatively open subset of a half-space, because most hypersurfaces in $\mathbb C^n$ are not biholomorphically equivalent to a plane such as $\{z: \operatorname{Im} z=0\}$.