Find all non zero complex numbers z satisfying `barz=iz^2`

Answer

`[z =I , pm (sqrt3)/(2) - (i)/(2)]`

Explanation

Let z= x+iy.
Given `bar z= iz^2`
`rArr bar(x+ iy)=i(x+I y)^2`
`rArr x-iy =i(x^2-y^2+2i xy)`
`rArr x- iy = -2 xy +I ( x^2 -y^2 +2i xy)`
`rArr x-iy= -2 xy +i(x^2 -y^2)`
Note it is a compound equation , therefore we can generate from it more than one primary equation .
On equationg real and imaginary parts equations.
x= -2 xy and `x^2-y^2+y=0`
`rArr x(1 +2y)=0`
`rArr` x=0 or y = -1/2
When y=- 1/2 , `x^2-y^2+y=0`
`rArr x^2 -1/4-1/2=0 rarr x^2 = 3/4`
`rArr x=pm (sqrt3)/(2)`
Therefore z=0 +i 0, 0+ i , `pm sqrt(3)/2-i/2`
`rArr z=i , pm sqrt(3)/2-i/2 " " [ because z ne 0]`