  We apply Wildeshaus's theory of motivic intermediate extensions to the
motivic decomposition conjecture, formulated by Deninger-Murre and
Corti-Hanamura. We first obtain a general motivic decomposition for the Chow
motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose
geometric fibers are Tate. Using Voevodsky's motives with rational
coefficients, the formula is valid for an arbitrary regular base $S$, without
assuming the existence of a base field or even of a prime integer $\ell$
invertible on $S$. This result, and some of Bondarko' ideas, lead us to a
generalized formulation of Corti-Hanamura's conjecture. Secondly we establish
the existence of the motivic decomposition when $f:X \rightarrow S$ is a
projective quadric bundle over a characteristic $0$ base, which is either
sufficiently general or whose discriminant locus is a normal crossing divisor.
This provides a motivic lift of the Bernstein-Beilinson-Deligne decomposition
in this setting.
