I will only add the note that one can also use ToRules[] to get the results of Reduce[] in a more usable form:

Simplify[ToRadicals[z /. {ToRules[Reduce[z^3 - 2 Abs[z] + 1 == 0 && Abs[Im[z]] > 0, z]]}]]
   {-(1/2) I (-I + Sqrt[3]), 1/2 I (I + Sqrt[3]), 
    1/4 (1 - Sqrt[5] - I Sqrt[18 - 6 Sqrt[5]]),
    1/4 (1 - Sqrt[5] + I Sqrt[18 - 6 Sqrt[5]]),
    (1/(2 6^(2/3)))(-4 (3/(9 + Sqrt[177]))^(1/3) + (2 (9 + Sqrt[177]))^(1/3) -
    I 6^(1/6) Sqrt[-24 + 24 3^(1/3) (2/(9 + Sqrt[177]))^(2/3) + 2^(1/3) (3 (9 + Sqrt[177]))^(2/3)]),
    (1/(2 6^(2/3)))(-4 (3/(9 + Sqrt[177]))^(1/3) + (2 (9 + Sqrt[177]))^(1/3) +
    I 6^(1/6) Sqrt[-24 + 24 3^(1/3) (2/(9 + Sqrt[177]))^(2/3) + 2^(1/3) (3 (9 + Sqrt[177]))^(2/3)])}

where I have also elected to use ToRadicals[] to see the solutions in radical form.