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.