Burgundy can set some fruit alone, but it does better with a partner. Plums are picky about pollination, so matching the right plum group is important. It flowers in bloom group 3, which is mid bloom, so a partner has to be flowering at the same time to be any use. Usually grown in USDA zones 6-9.
No single variety stands out as the obvious partner, but that is not a problem here: Elephant Heart, Santa Rosa, Satsuma, Beauty, and Hollywood all bloom close enough to work. There are 2 more in the list below. Pick whichever your nursery actually carries. Keep partner trees within about 50 feet if you can. Plum bloom is early, so having a well-matched partner improves the odds in cool spring weather.
One thing to keep in mind. Do not casually mix European, Japanese, and hardy hybrid plums. Plum group matters.
To put that into practice: at a minimum, plant two compatible plums from the same pollination group, such as European with European or Japanese with Japanese. Better still, use two or three plums in the same group with overlapping bloom so frost, poor weather, or shy bee activity does not ruin fruit set. If you only have room for one plum, choose a self-fruitful European type, but still plan for a second compatible plum later if yield matters.