Seems strange, when I simply copy and paste your code, i get
julia> sol = solve(prob, Vern7(), dt = 0.1, adaptive = false, reltol=1e-8);
julia> sol(1)
2-element Vector{ComplexF64}:
3.125989776501887 + 0.000000000000000im
5.058513100378909 + 0.000000000000000im
julia>
which corresponds well enough to your analytical solution