Some artificial chemistries model the synthesis, evolution and complexity of digital life consisting of sequences of computer operations (opcodes) that are driven by point mutations to compete for memory and CPU time. One of us previously built Amoeba, a computer world inspired by Tierra and designed to study the emergence of self-replicating sequences of opcodes from a prebiotic world initially populated by randomly selected opcodes. Eventually an ancestral opcode sequence would emerge. The current version of Amoeba uses a computationally universal opcode basis set and the same addressing as Tierra. It was previously thought such changes would preclude the emergence of self-replicators. Instead, these modifications radically affect the emergence of self-replicators from the primordial soup; Amoeba exhibits a self-organization phase, after which self-replicators emerge. First, the opcode basis set becomes biased. Second, short opcode building blocks are propagated throughout memory space. When sufficiently dense, these prebiotic sequences combine to form self-replicators. Self-organization is quantified by measuring the time evolution of n-opcode sequence frequencies, the size distribution of sequences, and the mutual information of opcode pairs.