  We explore the applicability of splitting methods involving complex
coefficients to solve numerically the time-dependent Schr\"odinger equation. We
prove that a particular class of integrators are conjugate to unitary methods
for sufficiently small step sizes when applied to problems defined in the group
$\mathrm{SU}(2)$. In the general case, the error in both the energy and the
norm of the numerical approximation provided by these methods does not possess
a secular component over long time intervals, when combined with
pseudo-spectral discretization techniques in space.
