Abstract:  The classical Illumination number of a convex body $K$ is the minimal number of light sources that can illuminate the boundary of $K$. The weighted illumination number is the infimal total mass of an illuminating weighted set of directions. The famous Hadwiger Conjecture states that among all convex bodies in $R^n$, the cube has the largest illumination number, both classical and weighted. In this talk we will discuss a Hadwiger-type conjecture for complex convex bodies $K$ in $C^n$. Here the role of the cube will be played by the polydisc, that is, the product of $2$-dimensional discs. First, we will discuss the illumination number of the polydisc, relating the problem to a covering problem for the torus. Then we will sketch a proof of the complex Hadwiger conjecture for complex zonotopes and complex zonoids. This talk is based on joint work with Liran Rotem and Boaz Slomka.