Related, but not exactly the same, to this question: "Should I rewrite my question, or create a new one?"
Recently I gave a "quick and dirty" answer (in fact I just googled for getting it) to this interesting question. The asker reviewed it and commented very politely on why it misses to answer the question, without downvoting (or insulting me, which perhaps it would have been the case...).
After that, I remembered some interesting facts on simple iterations and fixed point of holomorphic functions in a single variable, and worked out an entirely different answer. Meanwhile another user gave the question a very interesting answer, and they cite the paper I cited: so, quoting exactly the title of this question,
Should I give a new answer or change the old one?
Note that my answer was not downvoted nor upvoted, so there would be no harm in deleting it and, for example, adding its reference to the other answer.