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

Answer

`z = I, pm (sqrt(3))/(2)-(i)/(2)`

Explanation

Let `z = x +iy`. Then
`barz = iz^(2)`
`rArr x -iy = i(x^(2) -y^(2)+2ixy)`
`rArr x -iy = i(x^(2) -y^(2))-2xy`
`rArr x(1+2y)=0" "(1)`
`and x^(2) -y^(2) + y =0 " "(2)`
From (1),x= 0 or y = - 1/2. From (2), when x =0, y=0, 1 and
when `y = - 1//2`, `x = pm (sqrt(3//2)`. For non zero complex number z,
`z= i,(sqrt(3))/(2)-(i)/(2),(sqrt(3))/(2)-(i)/(2)`