Ice clouds play an important role in the atmospheric radiation budget, both by reflecting shortwave radiation and absorbing or emitting longwave radiation. These effects can modulate the cloud radiative heating (CRH) rate, which in turn influences circulation and precipitation. Ice cloud radiative properties depend on the size, shape (or habit), and complexity, including surface roughness or hollowness, of in-cloud ice crystals. To better predict ice-cloud radiative effects, there has been a continuous effort to account for more ice crystal habits and complexity in current radiative transfer calculations. Here, we conduct a series of idealized single-column radiative transfer calculations to study how ice-CRH responds to including ice crystal complexity. We evaluate four ice optical schemes for a range of ice cloud formation temperatures or altitudes, geometrical depths, ice water paths (IWP), and ice crystal effective radii. In addition, we present a heating rate sensitivity matrix as a condensed visualization of the CRH response across a broad parameter space. We find that including ice complexity in cold thin clouds with high IWP, can diminish the net in-cloud heating and cloud-top cooling by 2.5 K d^-1 and 15 K d^-1, respectively. Furthermore, while temperature-based schemes behave similarly to other schemes at warmer temperatures, they predict net CRH at cloud bottom more than 10 K d^-1 higher than size-dependent schemes at the coldest temperatures. Either weakening of CRH by ice complexity or strengthening by temperature-dependent schemes can alter anvil cloud lifetime and evolution, as well as large-scale atmospheric circulation.