http://web2.0calc.com/questions/what-is-square-root-of-i#r4

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Here is a direct algebraic answer:

 

Suppose that z=c+di, and we want to find √z=a+bi lying in the first two quadrants. So what are a and b?

 

Precisely we have:
a=Sqrt[c+sqrt(c^2+d^2)/2]

 

and

b=d/|d|Sqrt[-c+sqrt(c^2+d^2)/2]
(The factor of d/|d| is used so that b has the same sign as d) To find this, we can use brute force and the quadratic formula. Squaring, we would need to solve:
a^2−b^2+2abi=c+di.
This gives two equations and two unknowns (seperate into real and imaginary parts) which can then be solved by substitutions and the quadratic formula.

 

Hope that helps.