Answer in the projective version of the Klein model, cf. the Wikipedia article: hyperbolic $n$-space $\mathbb{H}^n$ is realized as the domain $U^n\subseteq \mathbb{RP}^n$ given by $U^n=\{[x_0:\cdots:x_n]\in\mathbb{RP}^n\mid Q(x_0,\dots,x_n)>0\}$ with $Q(x_0,\dots,x_n)=x_0^2-x_1^2-\cdots-x_n^2$. Geodesics are induced from straight lines in the projective space, and so faces of simplices are given by real hyperplanes in $\mathbb{RP}^n$.

Now complexification gives you: the ambient $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the quadric $Q(x_0,\dots,x_n)=0$ which is the boundary of $U^n$ becomes the quadric $Q$ mentioned by Goncharov, and the real hyperplanes giving the faces of the simplex become the complex hyperplanes.

Answer in the ball version of the Klein model: hyperbolic $n$-space is realized as $\{(x_1,\dots,x_n)\in \mathbb{R}^n\mid \sum x_i^2<1\}$. This can be compactified via $\mathbb{R}^n\subseteq\mathbb{RP}^n$. Complexification gives you the same thing as above: $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the boundary sphere $S^n\subseteq\mathbb{R}^n$ becomes the complex quadric hypersurface defined by $\sum x_i^2-1=0$, and the faces of the simplices give you complex hyperplanes.

Both versions are related by compactification and coordinate change.

Answer in the projective version of the Klein model, cf. the Wikipedia article: hyperbolic $n$-space $\mathbb{H}^n$ is realized as the domain $U^n\subseteq \mathbb{RP}^n$ given by $U^n=\{[x_0:\cdots:x_n]\in\mathbb{RP}^n\mid Q(x_0,\dots,x_n)>0\}$ with $Q(x_0,\dots,x_n)=x_0^2-x_1^2-\cdots-x_n^2$. Geodesics are induced from straight lines in the projective space, and so faces of simplices are given by real hyperplanes in $\mathbb{RP}^n$.

Now complexification gives you: the ambient $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the quadric $Q(x_0,\dots,x_n)=0$ which is the boundary of $U^n$ becomes the quadric $Q$ mentioned by Goncharov, and the real hyperplanes giving the faces of the simplex become the complex hyperplanes.

Answer in the ball version of the Klein model: hyperbolic $n$-space is realized as $\{(x_1,\dots,x_n)\in \mathbb{R}^n\mid \sum x_i^2<1\}$. This can be compactified via $\mathbb{R}^n\subseteq\mathbb{RP}^n$. Complexification gives you the same thing as above: $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the boundary sphere $S^n\subseteq\mathbb{R}^n$ becomes the complex quadric hypersurface defined by $\sum x_i^2-1=0$, and the faces of the simplices give you complex hyperplanes.

Both versions are related by compactification and coordinate change.

Maybe I should also point out that the quadric in the complex case is relevant data to remember: from the quadric and the hyperplanes you can recover the vertices of the simplex.

I think the quadric is easier to see in the projective version of the Klein model, cf. the Wikipedia article: hyperbolic $n$-space $\mathbb{H}^n$ is realized as the domain $U^n\subseteq \mathbb{RP}^n$ given by $U^n=\{[x_0:\cdots:x_n]\in\mathbb{RP}^n\mid Q(x_0,\dots,x_n)>0\}$ with $Q(x_0,\dots,x_n)=x_0^2-x_1^2-\cdots-x_n^2$. Geodesics are induced from straight lines in the projective space, and so faces of simplices are given by real hyperplanes in $\mathbb{RP}^n$.

Now complexification gives you: the ambient $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the quadric $Q(x_0,\dots,x_n)=0$ which is the boundary of $U^n$ becomes the quadric $Q$ mentioned by Goncharov, and the real hyperplanes giving the faces of the simplex become the complex hyperplanes.

Finally, the projective version of the Klein model is related to the disk version of the Klein model via compactification (and coordinate change). So, in the disk model, the quadric $Q$ is the complexification of the boundary sphere of the open unit ball.

Answer in the projective version of the Klein model, cf. the Wikipedia article: hyperbolic $n$-space $\mathbb{H}^n$ is realized as the domain $U^n\subseteq \mathbb{RP}^n$ given by $U^n=\{[x_0:\cdots:x_n]\in\mathbb{RP}^n\mid Q(x_0,\dots,x_n)>0\}$ with $Q(x_0,\dots,x_n)=x_0^2-x_1^2-\cdots-x_n^2$. Geodesics are induced from straight lines in the projective space, and so faces of simplices are given by real hyperplanes in $\mathbb{RP}^n$.

Now complexification gives you: the ambient $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the quadric $Q(x_0,\dots,x_n)=0$ which is the boundary of $U^n$ becomes the quadric $Q$ mentioned by Goncharov, and the real hyperplanes giving the faces of the simplex become the complex hyperplanes.

Answer in the ball version of the Klein model: hyperbolic $n$-space is realized as $\{(x_1,\dots,x_n)\in \mathbb{R}^n\mid \sum x_i^2<1\}$. This can be compactified via $\mathbb{R}^n\subseteq\mathbb{RP}^n$. Complexification gives you the same thing as above: $\mathbb{RP}^n$ becomes $\mathbb{CP}^n$, the boundary sphere $S^n\subseteq\mathbb{R}^n$ becomes the complex quadric hypersurface defined by $\sum x_i^2-1=0$, and the faces of the simplices give you complex hyperplanes.

Both versions are related by compactification and coordinate change.