 Brownian recurrence proof of the fundamental theorem of algebra

If a nonconstant complex polynomial were zero-free, its reciprocal would be an entire bounded function. The real part of its value along planar Brownian motion is a bounded martingale, hence converges almost surely. If that real part were nonconstant, continuity would give two disks with separated value ranges, and recurrence of planar Brownian motion would force alternate visits to both at arbitrarily large times, contradicting convergence. It is therefore constant; the Cauchy-Riemann equations then make the reciprocal constant, a contradiction.