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.