There are many synonyms for the phrase "argue animatedly," each with a slightly different connotation. One might "debate with enthusiasm," "quarrel passionately," "contest fiercely," or "dispute ardently." Other options include "disagree vehemently," "wrangle intensely," "spar energetically," or "spar heatedly." Regardless of the phrasing, this type of communication involves a great deal of emotion and energy. Those who argue animatedly may be deeply invested in their beliefs, or may simply enjoy the thrill of a lively debate. In any case, their passion and intensity are sure to get their point across.