Question 857211
This may be only part of the answer, but {{{x^2=i}}} if {{{x=cos(pi/4)+i*sin(pi/4)}}}.


How? 
Draw a unit circle.  The horizontal axis is Real and the vertical axis is Imaginary.  Horizontal intercepts are 1 and -1.  Vertical intercepts are i and -i.  Think of multiplications by whole number powers of i to be rotations starting at (1,0).  If 1*1, get 1.  If 1*-1, get -1.  If 1*i, get i.  If 1*(i)(i), get same as 1*(-1) which is -1.  Thinking this way, the way to go from negative 1 to HALF WAY from negative 1 to positive 1 is the take square root of -1.  


In that same way, if you start with {{{x^2=i}}}, and you want what is x, this is like starting at (0,i) on this complex unit circle, and rotating half-way from (0,i) to (1,0).  This puts your point on an angle of positive {{{pi/4}}}.  The coordinates on this point are as (cos(pi/4),i*sin(pi/4)), which you can represent as {{{highlight(cos(pi/4)+i*sin(pi/4))}}}.