These are native to central/northern China, Japan, and Korea. In the springtime you can find them at any big box store or nursery. The challenge is finding one with a single trunk or just two or three. Most of what is sold is a thicket of small trunks. I purchased this from a Home Depot in 2012. Mike Horine noticed they had several with single thick trunks.
This first picture is from Feb 2021. It is ready for the new growing season. I keep it in the garage over the winter. You can't tell from the picture, but the buds are starting to swell. Soon I will be hauling it into the sun each day and back into the garage at night. The substrate mix is equal parts Akadama, Lava, Pumice, and Organic.
This is the only burning bush I have kept so everything I think is a case of one. A couple of observations.
I have had no success cutting back a larger branch to before any secondary branches. I am surprised by this since it is basically a bush that people shear in the landscape. You can see one I did in 2019 in the crotch 2/3 of the way up on the left. I now only cut back to just after a secondary branch.
It does bud back well on the trunk, you can see a couple about mid way up, and older branches. So for the longer branches I need to shorten I wait and hope to see something bud out.
At the point of the cut it will push out lots of small branches. These need to be reduced to two that year otherwise the cut end will turn into a large knob.
I have heard from a very reputable source that these can be pruned back to one bud pair in early summer after they have hardened off and they will push out a second flush. I tried that last year. It wasn't very successful, only about 10% second flush. I think my mistake was that I repotted it in March last year and I should have let it go for a year to recover.
The plan for this year is to let it grow and strengthen all summer, minimal fertilizer in the spring and then a bit more into the fall to prepare for 2022.
So far it hasn't really earned the name burning bush. The fall color tends to be a brownish red instead of the flaming red they are known for. Perhaps this year.
Below is a picture from March 2017. Notice the differences in the new twigs which are green, the ones a bit older which still have the green but are starting to have brown ribs along the branch, and the oldest where it is completely barked over like the trunk.
I have a small one and have been experimenting as well @dlmcpeters . It seems that if it gets plenty of sun I end up with better leaf color, but could be a genetic component as well.
I was able to do a partial outer canopy defoliation last year, and get lots of back budding. Then once it was dormant I was able to cut back to smaller/finer branching and buds. I also try to remove excess shoots. I think this is how I will continue to attempt development on it for now.
Nice tree, very nice actually. I had a number of very large trunked BB right up to that big freeze came and went. 70 degrees to -15 in 24 hours. That killed all of them except one. All were in gravel bunkers. I have a number of these and I cut them back all the time. I did put one in a bonsai pot last spring. It was not happy with me at all. Not much growth on it all year, but it was still green at the end of the year. Hopefully it will grow better this year. I purchased one at a club auction which is very nice and it grows pretty good. I trim it back to and it turns red, its ready for a pot. A couple smaller single trunk ones seem to get leggy and they are going to get chopped backed this spring. Another large potted one has been chopped back several times and it about ready for a pot. The last surviving big trunked one has grown down into the squeegee gravel and is going to be evicted from the bunker this spring. It is interesting that tree has very fine roots when growing in the squeegee. I do mean very fine roots. I have trimmed it back some because it is to tall and sticks out into the walkway. So far I have not killed any personally which is good, so they must be pretty tough. I do use Plant Tone on them and give them some of the Azalea fertilizer I like. All of these turn bright red. I wonder if you bring it into the garage to soon? Mine stay out in the cold until the bitter end before bringing them into the garage. I have noticed on other trees they do not color up as much when I run them into and out of the garage in the fall. I probably should just leave them outside and take my chances. But then, I don't care about the color at this point. Cutting branches back is no guarantee they will bud out where you want them. Cutting Alders back is a crap shoot on them. You can bet it is going to die well back on the trunk. I like the trunk on your tree and the nebari is great. Larry Jackel has a small one complete with deadwood at his house. I think it is a nice tree and time for a good Bonsai pot.