How mitochondria shaped the evolution of eukaryotic complexity has been controversial for decades. The discovery of the Asgard archaea, which harbor close phylogenetic ties to the eukaryotes, supports the idea that a critical endosymbiosis between an archaeal host and a bacterial endosymbiont transformed the selective constraints present at the origin of eukaryotes. Cultured Asgard archaea are typically prokaryotic in both size and internal morphology, albeit featuring extensive protrusions. The acquisition of the mitochondrial predecessor by an archaeal host cell fundamentally altered the topology of genes in relation to bioenergetic membranes. Mitochondria internalised not only the bioenergetic membranes but also the genetic machinery needed for local control of oxidative phosphorylation. Gene loss from mitochondria enabled expansion of the nuclear genome, giving rise to an extreme genomic asymmetry that is ancestral to all extant eukaryotes. This genomic restructuring gave eukaryotes thousands of fold more energy availability per gene. In principle, that difference can support more and larger genes, far more non-coding DNA, greater regulatory complexity, and thousands of fold more protein synthesis per gene. These changes released eukaryotes from the bioenergetic constraints on prokaryotes, facilitating the evolution of morphological complexity.