The companion matrix C of a (monic) polynomial p is uniquely associated with the representation of p in terms of monomials, and p is the characteristic polynomial of C. An equivalence transformation of C corresponds to a different representation of p, for example in a (generalized) Newton form. In the literature, this basic fact has been used for various algorithmic purposes, e.g. for the refinement of given approximations of the zeros of p using tools from Numerical Linear Algebra. In this talk we take a different point of view on companion matrices. In Numerical Analysis, for instance, discretization methods are often characterized by certain polynomials resp. their companion matrices C. A typical example are linear multistep methods for ODEs. In this way, theoretical questions like stability issues can be formulated in terms of properties of C. In this talk we demonstrate that the successs of this approach crucially depends on the choice of basis resp. an appropriate canonical form for C. Important special cases are the Bidiagonal and the Bidiagonal-Frobenius form which are associated with generalized Newton representations for p. In contrast to the Jordan form, these canonical forms are typically continuous w.r.t. certain parameters of the problem or the method considered, which make them a very useful and flexible tool for theoretical purposes, e.g. in stability theory. Some simple examples for this type of analysis will be given.