The effect you're seeing is probably a combination of chromatic aberration and spherical aberration. Both exist to some extent in all lenses, but different lenses vary greatly in how susceptible they are to these effects. Not surprisingly, more expensive lenses are usually more resistant to such optical imperfections, containing lens elements specifically designed to counteract them.
Chromatic aberration is caused by dispersion, the fact that different wavelengths of light have subtly different refractive indices, as seen in a rainbow. It usually manifests as coloured "fringing" around objects in a picture, particularly near the corners. This might be the "cyan-red glasses" effect you mentioned. In high-quality lens designs, it is combated with lens elements partially canceling out another element's dispersion, as well as using special low-dispersion materials such as fluorite glass in critical elements.
Spherical aberration affects the bokeh - the "evenness" or "smoothness" of the blur in out-of-focus areas of the image. In an ideal world, the out-of-focus image of a point would be an evenly-shaded disc. In reality, however, the discs look more like the ones in this image. In particular, if the lens has positive spherical aberration, the perimeter of the bokeh discs of objects behind the focal plane appears brighter than the center. Now, if the image contains long and thin details, such as grass or thin branches, you might see how this may cause the sort of "duplication" you mentioned. In lens design, spherical aberration is typically compensated for by using one or more nonspherical lens elements. Note that negative spherical aberration in a lens design may be desirable, as it may result in smoother, more pleasing background bokeh.