This paper explores the electrical and thermal conductivity of complex contact spots on the surface of a half-space. Employing an in-house Fast Boundary Element Method implementation, various complex geometries were studied. Our investigation begins with annulus contact spots to assess the impact of connectedness. We then study shape effects on "multi-petal" spots exhibiting dihedral symmetry, resembling flowers, stars, and gears. The analysis culminates with self-affine shapes, representing a multiscale generalization of the multi-petal forms. In each case, we introduce appropriate normalizations and develop phenomenological models. For multi-petal shapes, our model relies on a single geometric parameter: the normalized number of "petals". This approach inspired the form of the phenomenological model for self-affine spots, which maintains physical consistency and relies on four geometric characteristics: standard deviation, second spectral moment, Nayak parameter, and Hurst exponent. As a by-product, these models enabled us to suggest flux estimations for an infinite number of petals and the fractal limit. This study represents an initial step into understanding the conductivity of complex contact interfaces, which commonly occur in the contact of rough surfaces.