Let $f,g\in C^\infty(\mathbb R^n;\mathbb R)$ be two Morse functions having both a critical point at $0$. Is it always possible to find local coordinates around $0$ such that both $f $ and $g$ become quadratic in the new coordinates?
After the comment of Matt, I realized that i forgot an important assumption: $0$ is a critical point of index $0$ of $f$.