  This White Paper is an input to the ongoing discussion about the extension
and refinement of simplified Dark Matter (DM) models. Based on two concrete
examples, we show how existing simplified DM models (SDMM) can be extended to
provide a more accurate and comprehensive framework to interpret and
characterise collider searches. In the first example we extend the canonical
SDMM with a scalar mediator to include mixing with the Higgs boson. We show
that this approach not only provides a better description of the underlying
kinematic properties that a complete model would possess, but also offers the
option of using this more realistic class of scalar mixing models to compare
and combine consistently searches based on different experimental signatures.
The second example outlines how a new physics signal observed in a visible
channel can be connected to DM by extending a simplified model including
effective couplings. This discovery scenario uses the recently observed excess
in the high-mass diphoton searches of ATLAS and CMS for a case study to show
that such a pragmatic approach can aid the experimental search programme to
verify/falsify a potential signal and to study its underlying nature. In the
next part of the White Paper we outline other interesting options for SDMM that
could be studied in more detail in the future. Finally, we discuss important
aspects of supersymmetric models for DM and how these could help to develop of
more complete SDMM.
