Here is a counterexample with bounded $\Omega$: take $m=2$, $\Omega=\{(z,w):|z|^2+|w|^2\leq R\}$, 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 $\{|z|=|w|=1\}$). 

Edit: this is not quite a counterexample you are looking for; I was trying to point out that even if extensions are possible, not all of them are pluriharmonic in the whole space. I am making my answer a CW until a more precise  one is found.