What if the patterns hidden within dialogue reveal more about communication than the words themselves? We introduce Conversational DNA, a novel visual language that treats any dialogue -- whether between humans, between human and AI, or among groups -- as a living system with interpretable structure that can be visualized, compared, and understood. Unlike traditional conversation analysis that reduces rich interaction to statistical summaries, our approach reveals the temporal architecture of dialogue through biological metaphors. Linguistic complexity flows through strand thickness, emotional trajectories cascade through color gradients, conversational relevance forms through connecting elements, and topic coherence maintains structural integrity through helical patterns. Through exploratory analysis of therapeutic conversations and historically significant human-AI dialogues, we demonstrate how this visualization approach reveals interaction patterns that traditional methods miss. Our work contributes a new creative framework for understanding communication that bridges data visualization, human-computer interaction, and the fundamental question of what makes dialogue meaningful in an age where humans increasingly converse with artificial minds.