In the irreducible case, $V$ has an invariant complex structure iff the character function has $L_2$-norm equal to $\sqrt2$ or $2$ (otherwise it is equal to $1$). 
In fact, $V\otimes \mathbb C=U\oplus \bar U$ 
for a complex representation $U$ and $\chi_V=\chi_{V\otimes\mathbb C}=\chi_U+\chi_{\bar U}$ where $\chi_U$ and $\chi_{\bar U}$ are orthogonal (unit) vectors
in case $U$ and $\bar U$ are inequivalent, or $U$ and $\bar U$ are equivalent and then $W$ admits even a quaternionic structure.