Luxury real estate investors prioritize personalization, location, privacy, quality finishes, and architectural design when purchasing a property. They want a home that meets their individual and family needs, is located away from noise in a secluded area with a large garden, has high quality durable materials like granite and slate, and an architectural design with additional amenities like a swimming pool or smart home technologies. Location is also important, as factors like proximity to work and amenities are a key consideration.