How do I find the square root of a complex number?

Say you want to find the square root of the complex number 3+2i.
We can assume that the answer we want will be in the form a+bi.
It follows then, that you can also write 3+2i as (a+bi)².
Expanding this gives us 3+2i = a²+2abi-b²
Then all we need to do is compare the coefficients of the imaginary and real parts: i.e. 3 = a²-b² and 2 = 2ab.
Solve these 2 simultaneous equations to get a =1.8 and b = 0.56 (ignore any imaginary solutions for a and b - they have to be real).
Therefore the square root of 3+2i is 1.8+0.56i. You can check this by squaring our solution and you'll get back to 3+2i (or near enough due to rounding).