We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded by a constant that depends only on the word. We describe how bounded additive complexity impacts the slope of the word and how a non-erasing morphism may affect the boundedness of a given words additive complexity. We prove the existence of recurrent words with constant additive complexity equal to any given odd positive integer. In the last section, we discuss a generalization of additive complexity. Our results suggest that words with bounded additive complexity may be viewed as a generalization of balanced words.