In this article, we use the complex compensated Horner's method to derive a compensated Ehrlich-Aberth method for the accurate computation of all roots of a polynomial. In particular, under suitable conditions, we prove that the limiting accuracy for the compensated Ehrlich-Aberth iterations is as accurate as if computed in twice the working precision and then rounded into the working precision. Moreover, we derive a running error bound for the complex compensated Horner's and use it to form robust stopping criteria for the compensated Ehrlich-Aberth iterations. Finally, extensive numerical experiments illustrate that the backward and forward errors of the root approximations computed via the compensated Ehrlich-Aberth method are similar to those obtained with a quadruple precision implementation of the Ehrlich-Aberth method with a significant speed-up in terms of the computation time.