Here is a counterexample with bounded $\Omega$: take $m=2$, $\Omega=\{(z,w):|z|\leq R, |w|\leq R\}$, where $R>1$, and $u(z,w)=\max\{\log^+|z|,\log^+|w|\}$.  Then $u$ is pluriharmonic on $\mathbb{C}^2\setminus \Omega$, but it is not a pluriharmonic function on $\mathbb{C}^2$ ($dd^cu\wedge dd^cu$ is a probability measure supported on $\{|z|=|w|=1\}$).