Reservoir computing (RC) is a computational framework in which a dynamical system, known as the reservoir, casts a temporal input signal to a high-dimensional space, and a trainable readout layer creates the output signal by extracting salient features from the reservoir. Several researchers have experimented with using the dynamical behavior of elementary cellular automaton (CA) rules as reservoirs. CA reservoirs have the potential to reduce the size, weight and power (SWaP) required to perform complex computation by orders of magnitude compared with traditional RC implementations. The research described in this paper expands this approach to CA rules with larger neighborhoods and/or more states, which are termed complex, as opposed to the elementary rules. Results show that some of these non-elementary cellular automaton rules outperform the best elementary rules at the standard benchmark five-bit memory task, requiring half the reservoir size to produce comparable results. This research is relevant to the design of simple, small, and low-power systems capable of performing complex computation.