In summary, complex numbers are defined as pairs of real numbers with specific rules for addition and multiplication. They cannot be ordered like real numbers, and the concept of "positive" and "negative" does not apply. The imaginary number, i, is defined as (0,1) and is the square root of -1 when multiplied by itself.
Of course, you can't DEFINE i as "square root of -1", not because -1 doesn't have a square root, but because, like any number, it has TWO.
Silly "paradoxes" like: i= [sqrt](-1) so i*i= i^{2}= [sqrt](-1)*[sqrt](-1)= [sqrt](-1*-1)= [sqrt](1)= 1 depend on that ambiguity.
When we are working in the real numbers, we can specify sqrt[x] as meaning the POSITVE root. In complex numbers, we don't have any way of distinguishing "positive" or "negative" (the complex numbers cannot be an ordered field).
The way complex numbers are properly defined is as PAIRS of real numbers (a,b) with addition defined as (a,b)+ (c,d)= (a+b, c+d) and multiplication defined as (a,b)*(c,d)= (ac-bd,ad+bc). It then follows that numbers of the form (a,0) act like real numbers while (0,1)*(0,1)= (0*0-1*1,0*1+1*0)= (-1,0). If we identify (0,1) with i (having dodged the question of how to distinguish between roots), we can write any complex number as (a, b)= a+ bi and have i*i= (-1,0)= -1.