Looking fixedly can be replaced by a number of synonyms to add variety to one's speech or writing. Some of the synonyms for looking fixedly include staring, gazing, peering, eyeballing, or staring intently. These terms describe a prolonged, intense gaze that may communicate interest, curiosity, or even intimidation. Other possible synonyms include examining closely, scrutinizing, inspecting, or surveying, which suggest a more deliberate or critical investigation of an object or subject. Alternately, one might use terms like ogling or leering to suggest a more inappropriate or lascivious type of gaze. Overall, there are numerous synonyms for looking fixedly that convey different shades of meaning and emotion, depending on context and intent.