We analyze a path-lifting algorithm for finding an approximate zero of a complex polynomial, and show that for any polynomial with distinct roots in the unit disk, the average number of iterates this algorithm requires is universally bounded by a constant times the log of the condition number. In particular, this bound is independent of the degree $d$ of the polynomial. The average is taken over initial values $z$ with $|z| = 1 + 1/d$ using uniform measure.