Recall the usual proof that commutative rings have maximal ideals: every chain of ideals has an upper bound (their union), and apply Zorn's. However, I always found this unsatisfying because there doesn't seem to be a relationship between the maximal ideal's existence and the union construction; it feels somewhat indirect. I find it cleaner to identify a maximal chain of ideals and directly construct the maximal ideal as the union. The claim that every poset has a maximal chain1 is called the Hausdorff Maximality Principle and fortunately it is almost trivially equal to Zorn's.
HMP $\imp$ Zorn's
Suppose every chain has an upper bound. By hypothesis there is a
maximal chain, and its upper bound is a maximal element.
Zorn's $\imp$ HMP
Consider the poset $P$ of chains under the subset relation. Every
chain in $P$ has an upper bound, which is its union; thus by Zorn's
there is a maximal element in $P$ which is a maximal chain.