With regard to the answer already provided:
The Arnold proof is well known to be erroneous, but a correct (as far as I know) version is cited in an earlier MO post here. In particular, it is a proof of the FTA via the Brouwer Fixed Point Theorem.
The latter source is:
Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.
Edit 1: Todd Trimble has kindly provided a link to the Fort paper that does not require JSTOR access.
Separately, I see the following quotation:
"Recently, there have been very interesting proofs of the Brouwer theorem. Kulpa deduced a generalization of the Brouwer theorem from the Fubini theorem and the Weierstrass approximation theorem, and applied it to give a simple proof of the fundamental theorem of algebra."
The source of this excerpt is:
Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF.
And the reference under discussion is:
W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.
I gathered this information at the request of D. Goroff some time ago, at which point my search for the Kulpa paper was, unfortunately, fruitless.
Edit 2: Karol Szumiło remarks that a friend in one of Warsaw's libraries was able to track down the Kulpa paper! Given the difficulty of finding it, I have uploaded a copy here.