Complex equations (find z that satisfy certain conditions)... I'd like to find all z in C (complex set) that satisfy : 

2*Im(z³) = 3*z*Im(z²) 

where Im(z) = imaginary part of z.
You are trying to solve $$2\operatorname{Im}(z^3)=3z\operatorname{Im}(z^2).$$
The left-hand side is real. So $3z\operatorname{Im}(z^2)$ must be real, which implies $z$ is real. Also, if $z$ is real then $\operatorname{Im}(z^3)=\operatorname{Im}(z^2) = 0$. So the equation is satisfied for all real $z$. As such, the solution is the set of real numbers, $\mathbb{R}$.
z^3= (a+ib)^3 = (a^3 - b^2 - 3ab^2) + i(3a^2b)

Im(z^3)= 3a^2b

2Im(z^3)=2(3a^2b)=6a^2b

then z^2 = (a+ib)^2 = (a^2 -b^2) +i2ab 

Im(z^2)= 2ab

3zIm(z^2)= 3(a+ib)2ab = (3a+i3b)(2ab)= 6a^2b+iab^2

Hence, 2Im(z^3)= 3zIm(z^2) 

= 6a^2b = 6a^2b+iab^2

which means that iab^2 = 0  so ab^2 =0 

the condition for the real and imaginary part of z =
Re(z) + 0Im(z) = Re(z)