Magnetic vortex is one of the simplest nontrivial topological structure that can exist in magnetic nanodots. Due to nonzero topological charge, vortices exhibit complex and interesting dynamic under external excitation, which includes excitation of translational (gyrotropic) and spin-wave modes. Here, the authors study the vortex dynamic in thick magnetic nanodots, when the vortex becomes essentially three dimensional. The third dimension is shown to lead to the appearance of additional spin-wave modes with characteristic curled structure (``curled'' modes), which becomes the lowest ones in frequency among spin-wave modes for a sufficiently thick nanodot. Besides an uncommon profile, the curled modes also show uncommon hybridization with gyrotropic modes: modes propagating in opposite directions (clockwise and counterclockwise) that can be hybridized with different order gyrotropic modes.