What’s “Complex” about complex cognition?

While prematurely committing to a conception of "complex cognition" promises to stifle inquiry, tentatively specifying and empirically determining the relevant senses of "complex" could very well serve to drive the research program forward. This strategy has proven particularly useful in debates around simplicity. However, given that "complex" is a multi-dimensional adjective, all-things-considered comparative evaluations of it might not be possible.