Conventional optical security devices provide authentication by manipulating a specific property of light to produce a distinctive optical signature. For instance, microscopic colour prints modulate the amplitude, whereas holograms typically modulate the phase of light. However, their relatively simple structure and behaviour is easily imitated. We designed a pixel that overlays a structural colour element onto a phase plate to control both the phase and amplitude of light, and arrayed these pixels into monolithic prints that exhibit complex behaviour. Our fabricated prints appear as colour images under white light, while projecting up to three different holograms under red, green, or blue laser illumination. These holographic colour prints are readily verified but challenging to emulate, and can provide enhanced security in anti-counterfeiting applications. As the prints encode information only in the surface relief of a single polymeric material, nanoscale 3D printing of customised masters may enable their mass-manufacture by nanoimprint lithography.