The human brain is characterised by idiosyncratic patterns of spontaneous thought, rendering each brain uniquely identifiable from its neural activity. However, deep general anaesthesia suppresses subjective experience. Does it also suppress what makes each brain unique? Here we used functional MRI under the effects of the general anaesthetics sevoflurane and propofol to determine whether anaesthetic-induced unconsciousness diminishes the uniqueness of the human brain: both with respect to the brains of other individuals, and the brains of another species. We report that under anaesthesia individual brains become less self-similar and less distinguishable from each other. Loss of distinctiveness is highly organised: it co-localises with the archetypal sensory-association axis, correlating with genetic and morphometric markers of phylogenetic differences between humans and other primates. This effect is more evident at greater anaesthetic depths, reproducible across sevoflurane and propofol, and reversed upon recovery. Providing convergent evidence, we show that under anaesthesia the functional connectivity of the human brain becomes more similar to the macaque brain. Finally, anaesthesia diminishes the match between spontaneous brain activity and meta-analytic brain patterns aggregated from the NeuroSynth engine. Collectively, the present results reveal that anaesthetised human brains are not only less distinguishable from each other, but also less distinguishable from the brains of other primates, with specifically human-expanded regions being the most affected by anaesthesia.