complex numbers - Can someone verify whether i am correct in saying that the square root of negative one SHOULD NOT exist, make sense, or equal "i" or +/-i?

I am sure you are aware of the dillema of 1=-1 when we assume √-1=i. To refresh your memory it goes like this:

1= √1= √((-1)(-1))= √-1√-1= i*i= -1.

We must realize that saying √-1=i is only half true, therefore not true. √-1= +/-i. This is because i×i=-1 and so does -i×-i. Therefore we need to change our above statement to:

1= √1= √((-1)(-1))= √-1√-1= +/-i*+/-i= (+/-)-1=(+/-)1.

But then, wait, now 1 is equal to +/-1. It is equal to both itself and its opposite all of the sudden. Doesnt this make absolutely no sense? I think √-x should equal the set of no numbers. What do all of the educated mathematicians say?