  We study the relationship between Iitaka fibrations and the conjecture on the
existence of complements, assuming the good minimal model conjecture. In one
direction, we show that the conjecture on the existence of complements implies
the effective log Iitaka fibration conjecture. As a consequence, the effective
log Iitaka fibration conjecture holds in dimension $3$. In the other direction,
for any Calabi-Yau type variety $X$ such that $-K_X$ is nef, we show that $X$
has an $n$-complement for some universal constant $n$ depending only on the
dimension of $X$ and two natural invariants of a general fiber of an Iitaka
fibration of $-K_X$. We also formulate the decomposable Iitaka fibration
conjecture, a variation of the effective log Iitaka fibration conjecture which
is closely related to the structure of ample models of pairs with non-rational
coefficients, and study its relationship with the forestated conjectures.
