Burgundy looks like a deep, rich wine. You want the effect you project to be one of cool sophistication, but at the same time, you don't want to put others off by being too carefully matched. Consider using complementary, analogous or neutral colours.
The classic colour wheel displays all colours (hues or true colours), tints (colours mixed with white) and shades (colours mixed with black). Looking at a colour wheel, you will see colours positioned opposite one another.
Burgundy is red darkened with black. You can find monochromatic burgundies and put these together with one another--as in a shirt, tie and colour square--the effect will be rich and understated. Pull everything together with a grey suit.
Red and violet are good colours to match with burgundy since they fall next to each other on the colour wheel. You can also match tints of red and violet as well as hues of blue with burgundy.
Burgundy matches well with hunter green (both are darker shades of red and green). Be careful about shades and tints of hunter green so you don't clash colours.
Outside the Box
When you are shopping for new clothing, look carefully in the store mirror to make sure the colours, tints and shades you are choosing work well with each other. Don't be afraid to match reds or violets with burgundies or to use analogous colours.