x ^ n is the exponentiation operation, not the root finder. If you want to find all the roots of a complex number, you can for example search for the roots of the corresponding polynomial using
PolynomialRoots.jl (but there are also other packages for looking for roots of polynomials, like
julia> using PolynomialRoots
julia> roots([8, 0, 0, -1])
The argument of
roots is the vector of coefficients of the polynomial, from the lowest to the highest one. This gives you the 3rd roots of 8.
In your case:
julia> roots([x, 0, 0, -1])
Note also that for the exponentiation operation
x ^ (1/3) with
x real you can use instead
cbrt(x), which is often faster and more accurate.