On the Bollob\'as theorem for complex $C(K)$-spaces

Let $K$ and $S$ be compact Hausdorff spaces. We prove a Bollob\'as-type theorem for operators between complex spaces of continuous functions. More precisely, every operator that almost attains its norm at an initial function can be approximated by a norm-attaining operator whose norm-attaining function remains close to the original one. Our proof combines the iterative method developed previously in the real case with the recent construction for complex measures. In particular, our result provides a quantitative strengthening of the complex version of the Johnson-Wolfe density theorem. As a byproduct of our construction, the same conclusion can be obtained when considering only compact operators.