I know for a complex vector bundle $E\to X$, we have Chern class $c_i(E)=(-1)^ic_i(E^*)$. Therefore, in many cases, $E\to X$ and $E^*\to X$ are not isomorphic.
I wonder if there exists some non-trivial complex vector bundle $E\to X$ s.t. $c_i(E)=0$ when $i$ is odd? Moreover under this condition, if there exists some non-trivial complex vector bundle $E\to X$ s.t., $E\to X$ and $E^*\to X$ are isomorphic?
As is shown here, there is a bijection between isomorphism classes of principal $SU(2)$-bundles over a four-dimensional manifold $X$ (or even CW complex) and $H^4(X; \mathbb{Z})$ given by $P \mapsto c_2(P)$.
So over any four-manifold $X$, and any choice of $c \in H^4(X; \mathbb{Z})$, there is a rank two complex vector bundle $E$ with $c_1(E) = 0$ and $c_2(E) = c$. Note that $c_1(E^*) = -c_1(E) = 0$ and $c_2(E^*) = c_2(E) = c$, so $E^* \cong E$. If $c \neq 0$, then $E\cong E^*$ is not trivial