Let $f: M \to R$ be a morse function from a closed n-manifold $M$ with a critical value $c$. Is it possible that there are two critical points $P_1$, $P_2$ with different index, but with the same critical value?
A possible counter example is below with $M$ diffeomorphic to the torus and $f$ the height function.
The simplest example is on $\mathbb{R}$. Try to investigate the function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=x(x^2-1)^2.$
Nice exercise: Why is degree 5 the lowest degree of a polynomial which is Morse and has critical points with same value but different index?