I am trying to find the example of an operator an operator $T\in L(V)$ where $V=C^{2}$.
Could this be an example?
Let $(e_{1},e_{2})$ be an orthonormal basis of $C^{2}$, an operator $T\in L(C^{2})$ such that $T(e_{1})=e_{1}$ and $T(e_{2})=ie_{2}$.
If not, can someone give me an example?
Your example seems to work. Recall that a linear map is normal if $TT^* = T^*T$. The matrix representation for the mapping you've defined is
$$ T \;\; =\;\; \left [ \begin{array}{cc} 1 & 0 \\ 0 & i \\ \end{array} \right ] $$
We can clearly compute that $TT^* = T^*T = I$, but
$$ T^* \;\; =\;\; \left [ \begin{array}{cc} 1 & 0 \\ 0 & -i \\ \end{array} \right ] \;\; \neq \;\; T. $$