Due to their many nice properties groups with automatic structure (automatic groups) have received a lot of attention in the literature. The multiplication of an automatic group can be realized through finite automata based on a regular set of (not necessarily unique) representatives for the group, and hence, each automatic group has a tractable word problem and low derivational complexity. Consequently it has been asked whether corresponding results also hold for monoids with automatic structure. Here we show that there exist finitely presented monoids with automatic structure that cannot be presented through finite and convergent string-rewriting systems, thus answering a question in the negative that is still open for the class of automatic groups. Secondly, we present an automatic monoid that has an exponential derivational complexity, which establishes another difference to the class of automatic groups. In fact, both our example monoids are bi-automatic. In addition, it follows from the first of our examples that a monoid which is given through a finite, noetherian, and weakly confluent string-rewriting system need not have finite derivation type.