What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? 
Is it 
$$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$
where $w^*$ denotes the complex conjugate?

I assume you mean the oriented area of that triangle (since the number changes sign when you swap $z_2$ and $z_3$). 
In order to verify it, allow me to write your formula as $$g(z_1,z_2,z_3)=\mathfrak{Im}\frac{\overline{z_1}(z_2-z_3)+z_3\overline{z_2}}{2}$$ It holds $$g(z_1+t,z_2+t,z_3+t)-g(z_1,z_2,z_3)= \mathfrak{Im}\frac{2\mathfrak{Re}(z_2\bar t)+\lvert t\rvert^2}{2}=0$$
So it is translation-invariant (hence, we need verify it only when $z_1=0$).
But that case is much easier.
