Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\sqrt{-i}= e^{-iπ/4}$ but $$i\sqrt{i} =e^{iπ/2}\cdot e^{iπ/4}=e^{3iπ/4}.$$
Thank you in advance!
No. There is an important distinction between $i = \sqrt{-1}$, which is wrong, and $i^2 = -1$, which is correct, namely that the wrong relation implies that you can use square roots.
The main thing to keep in mind is that the square root is two-valued. This gives some weird behaviour, and it is best to stay away from it entirely when complex numbers are involved. Especially $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails, like you've just noted. It does stay true if we allow "flexible" choices of square roots, for instance $$ 1 = \sqrt{1} = \sqrt{(-1)\cdot (-1)} = \sqrt{-1}\cdot \sqrt{-1} = i \cdot (-i) $$ where one of the square roots of $-1$ is chosen to be $i$, while the other is chosen to be $-i$. This is not a good habit to get into, and illustrates why you should at least be very careful with square roots when dealing with complex numbers.
