First you need to know or be able to derive:
j = e^{pi j/2 + 2 pi j n}, n=0, +/-1, ...
this is because: e^{pi j/2} = cos(pi/2) + j sin(pi/2) = j.
Now z^4 = j^3 = e^{3 pi j/2 + 6 pi j n}, and so:
z = e^{[3 pi j/2 + 6 pi j n]/4} = e^{3 pi j/8 + 3/2 pi j n},
Putting n =0, 1, 2, 3 should give four distinct values for z, and any
other value will again give one of these.
RonL