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Abstract Muon colliders are an exciting possibility for reaching the highest energies possible on the shortest timescale. They potentially combine the greatest strengths of e + e − and pp colliders by bridging the energy versus precision dichotomy. In this paper we study the sensitivity of Higgs properties that can be achieved with a future 3 or 10 TeV muon collider from single Higgs production. The results presented here represent the first comprehensive picture for the precision achievable including backgrounds and using fast detector simulation with Delphes. Additionally, we compare the results of fast detector simulation with available full simulation studies that include the muon collider specific Beam Induced Background, and show the results are largely unchanged. We comment on some of the strengths and weaknesses of a high energy muon collider for Higgs physics alone, and demonstrate the complementarity of such a collider with the LHC and e + e − Higgs factories. Furthermore, we discuss some of the exciting avenues for improving future results from both theoretical and detector R&D that could be undertaken.