Abstract: (joint with Johannes Schwab) In the first part, we will introduce Hitchin systems and motivate the study of complex Lagrangians through the Kapustin-Witten picture. Then we will restrict our attention to visible Lagrangians, where the Kapustin-Witten picture leads to a proposal of mirror dual hyperholomorphic subvarieties. In the second part, we give an existence criterion for visible Lagrangians in terms of certain half-translation structures on the underlying Riemann surface refereed to as pillowcase covers. We will explain this notion and comment on the proposed mirror dual subvarieties to these particular visible Lagrangians.