Why isn't set $\{ \langle i, j \rangle \mid W_i = \overline{W_j}\}$ a recursively enumerated set?
Hint: Use a reduction for the proof. If your  language $L=\{ \langle i, j \rangle \mid W_i = \overline W_j\}$ is r.e. then of course  every language where the second machine $M_j$ is fixed is also r.e.. Say, you pick for $M_j$ the Turing machine that always accepts, then you are left with showing that the language $L_0=\{ i \mid W_i = \emptyset \}$ is not r.e.. This again can be shown with a reduction (i.e., show that $\overline{\text{HALT}}\le_p  L_0$). Alternatively, show that $L_0$ is not decidable (Rice's Theorem), but $\overline{L_0}$ is Turing-recognizable.