I'm rather confused, Plato.
In the complex system, is \sqrt(1) means we want to find complex numbers z such that z^2=1?
That means in complex system, the \sqrt(1) = +1 and -1, doesn't it?
It differs from the real system that defines \sqrt(1) as a positive number x such that x^2=1. Thus the solution is just x=1, without x=-1?
CMIW
Plato, concerning \sqrt{-4}=\sqrt{4\cdot-1}=\sqrt{4}*\sqrt{-1}=2i.
But is it true when we want to find for example the solution of z^2-i*z+2=0, then we could use
z_12=(i \pm \sqrt((-i)^2-4*1*2))/2=(i \pm \sqrt(-9))/2=(i \pm \sqrt(9*(-1)))/2=(i \pm \sqrt(9)*\sqrt(-1))/2=2i or -i?
Absolutely NOT! Nowhere, is that true.
The fact is: the use of a radical is for real numbers.
Now it is true that for historical reasons we wave our hands and allow .
BUT the rules for radicals apply only to real numbers and do not carry over to complex.
Example: , (hand waving).
Because is not real, we do not apply rules for radicals.