We envision the concept of Thoughtful AI, a new human-AI interaction paradigm in which the AI behaves as a continuously thinking entity. Unlike conventional AI systems that operate on a turn-based, input-output model, Thoughtful AI autonomously generates, develops, and communicates its evolving thought process throughout an interaction. In this position paper, we argue that this thoughtfulness unlocks new possibilities for human-AI interaction by enabling proactive AI behavior, facilitating continuous cognitive alignment with users, and fostering more dynamic interaction experiences. We outline the conceptual foundations of Thoughtful AI, illustrate its potential through example projects, and envision how this paradigm can transform human-AI interaction in the future.