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.
