There is no 'ordinary' when it comes to AI. The human-AI experience is extraordinarily complex and specific to each person, yet dominant measures such as usability scales and engagement metrics flatten away nuance. We argue for AI phenomenology: a research stance that asks "How did it feel?" beyond the standard questions of "How well did it perform?" when interacting with AI systems. AI phenomenology acts as a paradigm for bidirectional human-AI alignment as it foregrounds users' first-person perceptions and interpretations of AI systems over time. We motivate AI phenomenology as a framework that captures how alignment is experienced, negotiated, and updated between users and AI systems. Tracing a lineage from Husserl through postphenomenology to Actor-Network Theory, and grounding our argument in three studies-two longitudinal studies with "Day", an AI companion, and a multi-method study of agentic AI in software engineering-we contribute a set of replicable methodological toolkits for conducting AI phenomenology research: instruments for capturing lived experience across personal and professional contexts, three design concepts (translucent design, agency-aware value alignment, temporal co-evolution tracking), and a concrete research agenda. We offer this toolkit not as a new paradigm but as a practical scaffold that researchers can adapt as AI systems-and the humans who live alongside them-continue to co-evolve.