Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group generated by $A$ and $B$ is free. Suppose furthermore that $A$ and $B$ do not preserve $\mathrm{h}$. My guess is that the following is true $$ \forall x \in \mathbb{C}^2 \ \exists g \in G \ \text{such that } \ \mathrm{h}(g(x)) > 0 $$
but I can't find a proof.
Edit : It is easy to see that the last statement is false : take two matrixes having one common eigenvector of negative norm. Those two matrixes will generically generate a free group. But I think if you had the hypothesis that $A$ and $B$ have no common eigenvector, the statement above holds. And I don't think in that case that no hypothesis on the algebraic properties of the group (other than the ones implied by this hypothesis) is required.
More generally I'm asking myself the following question : if $\mathrm{h}$ is an hermitian non degenerate form on $\mathbb{C}^n$ which is not definite positive nor definite negative. How "big" must be a group $G$ to send every $x \in \mathbb{C}^n$ on a vector of positive norm ?