Your students might find it useful to see this "visual approach" to proving the FTA:
Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach. Link.
For a more rigorous approach by the same author, see:
Velleman, D. J. (1997). Another proof of the fundamental theorem of algebra. Mathematics Magazine, 216-217. Link.
(The latter proof uses "the fact that entire functions can always be represented by power series.")
Finally, an entire list of proofs can be found here. I have not read closely through these, so I can neither make a more specific recommendation nor vouch for all of them; in particular:
Remark. In the list above, #60 "A topological proof of the fundamental theorem of algebra" (Arnold, 1949) is known to have errors. This is why he published a correction paper a couple years later (#58); the main idea, though, of using the Brouwer Fixed Point Theorem to prove the FTA has been carried out (though perhaps this is a result your post-Calc III students have not yet seen). In case this is useful, see:
Fort, M. K. (1952). Some properties of continuous functions. American Mathematical Monthly, 372-375. Link.
(The last two sources I became aware of while writing up an answer here.)
Unsurprisingly, there is a list of ways to prove the FTA on MathOverflow.
This is likely to include most any suggestion that could be posted here (on MESE) but the reverse inclusion is sure not to hold: The majority of these proofs are not written for students who just completed Calc III. On the other hand, three lectures with the FTA as a sole goal could cover a fair bit of material...