We present an improved search for axion-like polarization oscillations in the cosmic microwave background (CMB) with observations from the Keck Array. An all-sky, temporally sinusoidal rotation of CMB polarization, equivalent to a time-variable cosmic birefringence, is an observable manifestation of a local axion field and potentially allows a CMB polarimeter to detect axion-like dark matter directly. We describe improvements to the method presented in previous work, and we demonstrate the updated method with an expanded dataset consisting of the 2012-2015 observing seasons. We set limits on the axion-photon coupling constant for mass $m$ in the range $10^{-23}$-$10^{-18}~mathrm{eV}$, which corresponds to oscillation periods on the order of hours to years. Our results are consistent with the background model. For periods between $1$ and $30~mathrm{d}$ ($1.6 times 10^{-21} leq m leq 4.8 times 10^{-20}~mathrm{eV}$), the $95%$-confidence upper limits on rotation amplitude are approximately constant with a median of $0.27^circ$, which constrains the axion-photon coupling constant to $g_{phigamma} < (4.5 times 10^{-12}~mathrm{GeV}^{-1}) m/(10^{-21}~mathrm{eV}$), if axion-like particles constitute all of the dark matter. More than half of the collected BICEP dataset has yet to be analyzed, and several current and future CMB polarimetry experiments can apply the methods presented here to achieve comparable or superior constraints. In the coming years, oscillation measurements can achieve the sensitivity to rule out unexplored regions of the axion parameter space.
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