We computationally study periodic impedance-matched metal-dielectric metamaterials and the advantage of imprinting moth-eye surfaces on them. Impedance-matched metamaterials are known to act as strong, polarization-independent, broadband absorbers. However, in the infrared region far from the metal's plasma frequency, the reflection from metal layers dominates over the absorption. Using anti-reflective moth-eye surfaces we show that it is possible to obtain absorption independent of polarization or incidence angle, over an exceptionally broad frequency range from 400 nm to 6 μm.