Axiom of choice doesn't seem to make sense when talking about dense ordered sets, yet it is used as a justification for many statements in branches that range from measure theory to abstract algebra. Is there any substitute for Zorn's lemma or any other variant that doesn't require outlandish assumptions?

Links:
https://en.wikipedia.org/wiki/Zorn%27s_lemma#Equivalent_forms_of_Zorn's_lemma
http://diposit.ub.edu/dspace/bitstream/2445/121981/2/memoria.pdf
https://en.wikipedia.org/wiki/Axiom_of_choice