If you've ridden a city bus/train system, think about those times when there aren't any seats available and you end up standing. What's the best way to place your feet so you don't get thrown sideways by the inertia of the vehicle? The obvious answer is: not to put your feet in a straight line parallel or perpendicular to your hips or the direction of travel, i.e. you want to stagger them diagonally.
A few years back I went to a Physical Therapist for patellar tendonitis, and starting the 3rd week one of the exercises he had me do was the lunge stepping simultaneously out to the side and front (about 45 degrees as you say). I really liked the diagonal lunges, because the normal straight-ahead lunges seemed to:
A) pass more load/impact through the lower quads and the knee joint itself (which we want to avoid during the acute phase of an injury)
B) be more difficult to maintain balance when putting so much stress on a joint where I had some existing instability.
When doing the diagonal step-out lunges, it was easier for me to feel the muscles in my upper quads and glutes loading. It also gave me more natural bracing from the non-lunging leg and partially recruited the hip ab/ad-ductors for control through the entire Range of Motion of the exercise, which was very important early on when I would still occasionally get that sharp stabbing-ripping twinge of an overloaded patellar tendon.
After 6 weeks of PT I had so much strength and control I could do a lunge in pretty much any ROM geometrically possible. When I do leg workouts now, I'll sometimes do 'em straight, sometimes, diagonal, sometimes sideways. Personally, if I'm wanting to focus specifically on the look and development of the major leg muscles, I'll do straight lunges. If I want a little more focus on strengthening the whole lower body and improving motion/flexibility/stability for playing a sport, I'll do the diagonal or sideways lunges.