Recently, several papers have connected the affect of repeated differentiation on the roots of polynomials to free probability and random matrices. For polynomials with real roots, this connection can be expressed either as a sum of independent matrices or as the compression of a single matrix. The real rooted-ness of the polynomials allows for explicit connections between the compression of matrices and differentiation. However, these more explicit formulas are no longer available when the polynomial has complex roots. After briefly reviewing the real rooted case, we will discuss how to extend these ideas to the complex setting. With this connection to sums of random objects, we will then focus on a central limit theorem for repeated differentiation and polynomials which are stable with respect for differentiation. Based on joint work with Sean O'Rourke and David Renfrew.