  We provide a version for operators of the Bishop-Phelps-Bollob\'{a}s Theorem
when the domain space is the complex space $C_0(L)$. In fact we prove that the
pair $(C_0(L), Y)$ satisfies the Bishop-Phelps-Bollob\'{a}s property for
operators for every Hausdorff locally compact space $L$ and any
$\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y= L_p
(\mu)$ ($1 \le p < \infty $).
