Hi Anita!! I went and got out my color recipe book, hope I have something in it that will help you. Here are some ideas, maybe if you try pea-size mixes you'll see one that might work for you:
1 part Burnt Umber, 2 parts Fuchsia - I have this labeled in my book as "wine", so I'm not sure if you can get it to burgundy or not.
To get nearly to Alizarin Crimson, you might try Kato - 1 part Kato Red and 1 part Premo Violet.
Hmmm, don't have much for burgundy that doesn't use Alizarin... but I do have what I described in my note as "purple red", 1 part Cad Red & 1 part Ultra blue.
I hope some of this may help, or at least be a starting point for you. Wish I had more! I checked my Premo color chips but nothing I could describe as Burgundy. Good luck!