This paper examines the rising competition between computational and dynamic conceptualizations of complexity in economics. While computable economics views complexity as something rigorously defined based on concepts from probability, information, and computability criteria, dynamic complexity is based on whether a system endogenously and deterministically generates erratically dynamic behavior of certain kinds. One such behavior is the phenomenon of emergence, the appearance of new formes or structures at higher levels of a system from processes occurring at lower levels. While the two concepts can overlap, they represent substantially different perspectives. A competition of sorts between them my become more important as new, computerized market systems emerge and evolve to higher levels of complexity of both kinds.