Frequency shifts, radiative decay rates, the Ohmic loss contribution to the nonradiative decay rates, fluorescence yield, and photobleaching of a two-level atom radiating anywhere inside or outside a complex spherical nanoshell, i.e. a stratified sphere consisting of alternating silica and gold concentric spherical shells, are studied. The changes in the spectroscopic properties of an atom interacting with complex nanoshells are significantly enhanced, often more than two orders of magnitude, compared to the same atom interacting with a homogeneous dielectric sphere. The detected fluorescence intensity can be enhanced by 5 or more orders of magnitude. The changes strongly depend on the nanoshell parameters and the atom position. When an atom approaches a metal shell, decay rates are strongly enhanced yet fluorescence yield exhibits a well-known quenching. Rather contra-intuitively, the Ohmic loss contribution to the nonradiative decay rates for an atomic dipole within the silica core of larger nanoshells may be decreasing when the silica core -inner gold shell interface is approached. The quasistatic result that the radial frequency shift in a close proximity of a spherical shell interface is approximately twice as large as the tangential frequency shift appears to apply also for complex nanoshells. Significantly modified spectroscopic properties (see computer program freely available at http://www.wave-scattering.com) can be observed in a broad band comprising all (nonresonant) optical and near-infrared wavelengths.