The answer is negative. This easily follows from [[1], Theorem 1.1]. While the cited theorem addresses multiplicative functions with values in $\pm 1$, we may replace each $\pm i$ in the sequence $\sqrt\mu$ with $\pm 1$ and still get bounded partial sums if it had bounded partial sums to begin with. 


  [1]: https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/mtk.12117?casa_token=k9WHBy1EEMkAAAAA%3As8jrEq2inTwLvH83Arpckvi2mPnXnFhqLZ6iZ2kIZvA7nKBVABRzMxPjfjczEUw070chnhEuDWX7-Kk