In this talk, the notion of imaginary projections of polynomials and its connections to hyperbolic polynomials and their hyperbolicity cones will be discussed. This geometric view to study stability, more generally conic stability (defined on more general real cones) is motivated by the theory of amoebas as well as by the general goal to exploit convexity phenomena in algebraic geometry. It will be shown that for cones with a spectrahedral representation, certificates for the conic stability of a given polynomial turns into a semidefinite feasibility problem. This reduction to a semidefinite problem is based on techniques from the connection of containment of spectrahedra and positive maps. This talk is based on a joint work with Stephan Gardoll and Thorsten Theobald: Conic Stability of Polynomials and Positive maps.