  In this paper, we first introduce the notion of a product structure on a
$3$-Bihom-Lie algebra which is a Nijenhuis operator with some conditions. And
we provide that a $3$-Bihom-Lie algebra has a product structure if and only if
it is the direct sum of two vector spaces which are also Bihom subalgebras.
Then we give four special conditions such that each of them can make the
$3$-Bihom-Lie algebra has a special decomposition. Similarly, we introduce the
definition of a complex structure on a $3$-Bihom-Lie algebra and there are also
four types special complex structures. Finally, we show that the relation
between a complex structure and a product structure.
