>>73
z^2 = (a+i)^2 = a^2 -1 + 2a i
z^4 = { (a^2 -1) + 2ai}^2 = (a^2 -1)^2 -4a^2 + 4a(a^2 -1)i
が実数ということで

4a(a^2 -1) = 0
a = 0, ±1
a = 0のとき z^4 = i^4 = 1

a = ±1のとき
a^2 = 1
z^4 = -4