Hyperproperties have received increasing attention in the last decade due to their importance e.g. for security analyses. Past approaches have focussed on synchronous analyses, i.e. techniques in which different paths are compared lockstepwise. In this paper, we systematically study asynchronous analyses for hyperproperties by introducing both a novel automata model (Alternating Asynchronous Parity Automata) and the temporal fixpoint calculus Hμ, the first fixpoint calculus that can systematically express hyperproperties in an asynchronous manner and at the same time subsumes the existing logic HyperLTL. We show that the expressive power of both models coincides over fixed path assignments. The high expressive power of both models is evidenced by the fact that decision problems of interest are highly undecidable, i.e. not even arithmetical. As a remedy, we propose approximative analyses for both models that also induce natural decidable fragments.