Recurrent correlation neural networks (RCNNs), introduced by Chiueh and Goodman as an improved version of the bipolar correlation-based Hopfield neural network, can be used to implement high-capacity associative memories. In this paper, we extend the bipolar RCNNs for processing hypercomplex-valued data. Precisely, we present the mathematical background for a broad class of hypercomplex-valued RCNNs. Then, we provide the necessary conditions which ensure that a hypercomplex-valued RCNN always settles at an equilibrium using either synchronous or asynchronous update modes. Examples with bipolar, complex, hyperbolic, quaternion, and octonion-valued RCNNs are given to illustrate the theoretical results. Finally, computational experiments confirm the potential application of hypercomplex-valued RCNNs as associative memories designed for the storage and recall of gray-scale images.