Okay so warning I'm not used to american mathematics notations. So i work with complex numbers a lot and i noticed a little shortcut.
(1+i)^2= 1 +2i-1 = 2i
So we can say that
i= ( (1+i)÷sqrt (2) ) ^2
Sqrt (i) = (+or-)(1/sqrt (2) + ( 1/sqrt (2) ) i)
So sqrt (i)= -1/sqrt(2) - ( 1/sqrt(2)) i
Or sqrt (i) = 1/sqrt(2) + (1/sqrt(2)) i
Which is what mr blackpenredpen found.
Also, is this method correct? I'm still in high school so i may still be behind hahah. Also, big fan of your work.