The following is an explicitly defined complex line bundle $E\to\Sigma$ over a closed surface:
View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by (outward-pointing) unit normal vectors, $n(p)=(n_1(p),n_2(p),n_3(p))$. Given the Pauli matrices $\tau_1 =\bigl( \begin{smallmatrix}
0&i\\ i&0
\end{smallmatrix} \bigr)$ and $\tau_2=\bigl( \begin{smallmatrix}
0&-1\\ 1&0
\end{smallmatrix} \bigr)$ and $\tau_3=\bigl( \begin{smallmatrix}
i&0\\ 0&-i
\end{smallmatrix} \bigr)$, define $\tilde{n}:\Sigma\to M_2(\mathbb{C})$ by $p\mapsto \sum_{j=1}^3n_j(p)\tau_j$. Then $E$ is the subbundle of $\Sigma\times\mathbb{C}^2$ defined by $\lbrace (p,v)\;|\;\tilde{n}(p)v=iv\rbrace$.
Is $E$ familiar, say $E\cong T\Sigma$ or $E\cong\underline{\mathbb{C}}$?
This is an instance where brute force seems like the only way to approach these questions, as everything is written pointwise instead of being abstractly defined. But I don't have a brute force method at hand.
[Edit] I think I can show that $c_1(E)=-1$ for $\Sigma=S^2$, so that it's neither $TS^2$ nor $S^2\times\mathbb{C}$.