  We study interpretable sets in henselian and sigma-henselian valued fields
with value group elementarily equivalent to Q or Z. Our first result is an
Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely
ramified characteristic zero henselian fields -- relative to value group
imaginaries and residual linear imaginaries. We extend this result to the
valued difference context and show, in particular, that existentially closed
equicharacteristic zero multiplicative difference valued fields eliminate
imaginaries in the geometric sorts; the omega-increasing case corresponds to
the theory of the non-standard Frobenius automorphism acting on an
algebraically closed valued field.
  On the way, we establish some auxiliary results on separated pairs of
characteristic zero henselian fields and on imaginaries in linear structures
which are also of independent interest.
