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Yes, this is true. The point is that $\partial\bar\partial f = 0$ on the complement $C$ of a ball in $\mathbb{C}^n$ ($n\ge 2$) implies that $f = h_+ + \overline{h_-}$ for some functions $h_\pm$ that are holomorphic on $C$. (They are unique up to adding a constant to one and subtracting it from the other.) Now Hartogs' extension theorem implies that $h_\pm$ extend to be holomorphic on all of $\mathbb{C}^n$. Your boundedness assumption now says that the real and imaginary parts of $f$, which are harmonic, are bounded, which implies, by Liouville's Theorem, that they are constant.