This is a hard question as coaching is so personal. What one person looks for, another person might not like at all. But, I've made a coaching switch within the past couple of years, so I have a small perspective on what I look for. Take it with a grain of salt.
When I first started with old coach, I really liked her. I was coming back from 4 years off ice due to injury, and she had a style that helped me get my feet back under me. But I never felt like there was a long-term plan. There was no structure to the competition season, or pre-season. Advanced elements I just got frustrated on because the communication wasn't there anymore. I guess I feel like she was more of a teacher, where she was good at teaching the individual things.
My new coach is more of a coach versus a teacher. She's very good at teaching elements, but there's always a specific end game and everything builds on each other. If I'm having trouble with an element, she figures out an exercise that I can warm-up with in order to strengthen the underlying skills. Each competition season, we map out what we are doing, where we want to be at each point, and how we are going to get there. She and I work well together because we are both driven and she understands how my mind works. She treats me no differently than her competitive kids, in essence, and I appreciate that because that's how I'm wired. She also is constantly perfecting the details of everything, and makes sure that what I perform plays to my strengths while still challenging me at first.
The only way I figured all of this out, though, was by working with her as my jump coach first. The contrast between the two coaching styles became really apparent then. If I hadn't actively started working with new coach in that setting, I might honestly still be with old coach. Maybe not. But for me, looking forward to my lessons, to each new season, being able to communicate with my coach honestly, and feeling like I'm finally improving are all indications that this is a good match. She pushes me to try things I might be scared of and I trust her to know when I'm ready.
I think coaching relationships also evolve though. You can't guarantee that it will be a good long-term match as your skills as goals change. But if you can find someone you're comfortable with, can take constructive criticism from, and feel like you are improving, that's a pretty good start.