We report the results of a search for exotic decays of a Higgs boson with m_\mathrm{h}=125~\mathrm{GeV} to a pair of new light bosons, $\mathrm{a_1}$, where one of the light bosons decays to a pair of muons and the other one decays to a pair of b quarks. Such signatures are predicted in a number of well motivated extensions of the standard model, including the next-to-minimal supersymmetry and generic two Higgs doublet models with an additional scalar singlet. A data sample corresponding to an integrated luminosity of 19.7~\mathrm{fb}^{-1} recorded with the CMS detector in 2012 is exploited in $\mu^+\mu^-\mathrm{b}\overline{\mathrm{b}}$ final states where no statistically significant excess is observed with respect to the standard model backgrounds for different $\mathrm{m}_\mathrm{a_1}$ hypotheses above 25 GeV and below $\mathrm{m}_\mathrm{h}/2$. Upper limits are set on $\sigma_{\rm ggF}\times{\rm Br}(\mathrm{h}\to\mu^+\mu^-\mathrm{b}\overline{\mathrm{b}})$ and on the branching ratio itself with former ranging between 4 to 12~\mathrm{fb}, depending on the $\mathrm{m}_\mathrm{a_1}$ values. The limit on the branching ratio is ${\rm Br}(\mathrm{h}\to\mu^+\mu^-\mathrm{b}\overline{\mathrm{b}})<9\times 10^{-4}$ for the entire mass range.

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