A recent and very important contribution to the literature on the fundamental theorem of algebra is Joe Shipman's article "Improving the Fundamental Theorem of Algebra," Math. Intelligencer 29 (2007), 9-14. Here is one of his results: A field with the property that every polynomial whose degree is a prime number has a root is algebraically closed. This result is sharp in the sense that if any prime is omitted then the conclusion is false.
Shipman's paper should go a long way towards addressing Andrew L's question of whether there is a "purely algebraic proof" of the FTA. The above result of Shipman's shows that we can limit the topology/analysis to proving that every polynomial over $\mathbb{C}$ of prime degree has a root; the rest is pure algebra. If you wanted to try to limit the use of topology or analysis even further, then this part of the proof is where you should focus your attention.
