What is wrong with this complex circuit analysis?
The substitutions of \$I_1\$ and \$R_2\$ are incorrect.
 

\begin{cases}
  -V_x+I_2(j \omega L_1)=-I_1(R_2+j \omega L_1)
\\
\\
  0 +  I_2(R_1+\frac{1}{j \omega C_1}+\frac{1}{j \omega C_2}+j \omega L_1)=-I_1(j \omega L_1)
\end{cases}

\begin{cases}
  -V_x+2jI_2=\color{red}{-(-2)}(\color{red}{0.5}+2j)=1+4j
\\
\\
  0 +  (2-2j)I_2=\color{red}{-(-2)}I_1(2j)=4j
\end{cases}

The last part of your calculation is correct, but used incorrect values.
For completeness, I'll continue with the correct values.
Compared to your solution \$I_2\$ actually changes sign, so:
$$  I_2=-1+j$$
Substiting this into
$$-V_x+2jI_2=1+4j$$
yields
$$V_x=2j(-1+j)-1-4j=-3-6j$$