Let $G$ be a simple graph and $\overline{G}$ your complementary graph. If $G$ and $\overline{G}$ are both bipartite.
We say that $G$ is a simple graph $G=(X,U)$ if $X$ is a finite subset, nonempty and $U$ is a subset $X \otimes X=\{\{x,y\}\mid x,y\in X, x\neq y\}$
$\overline{G}$ is complementary graph of $G$, $\overline{G}=(X,\overline{U})$ with $\overline{U}=(X\otimes X)\backslash U$
We have that $G$ and your complementary $\overline{G}$ are both bipartite with the class of vertices $X_1$ and $X_2$ ($\{X_1,X_2\}$ is a partition of $X$, i.e., $X_1\neq \emptyset$, $X_2\neq \emptyset$, $X_1\cap X_2=\emptyset$, $X_1\cup X_2=X$) and none $U$ and $\overline{U}$ element has both extremities at vertices of the same class.
What can I conclude about $G$?
Any two vertices in $X_1$ have no edge in $G$, hence have an edge in $\overline G$. If $\overline G$ is bipartite, we conclude that $X_1$ (and similarly $X_2$) has at most two elements. This leaves only very finitely many possibilities for $G$.