LSU Purple can fruit alone, but it still benefits from a partner. Common figs do not need a second tree for pollination, but a second variety can still give you more total fruit and better backup. It flowers in summer fruiting season, which is worth knowing if you add a second tree later. Usually grown in USDA zones 7-10.
The good news is you have real choices here. The strongest partners are Black Mission, Brown Turkey, Chicago Hardy, and Violette de Bordeaux. Any one of them is enough to get fruit set going. If your nursery has none of those, Celeste, Desert King, or Kadota will also do the job. Figs do not need a nearby partner for normal home-garden fruiting, but grouping two varieties can still be handy for harvest timing and insurance.
One thing to keep in mind. Common figs do not need a second tree for pollination. Extra varieties are optional, not required. Chill note: Less than 300 chill hours.
To put that into practice: at a minimum, one common fig is enough for fruit in most home plantings. Better still, plant two common figs only if you want a longer season, more total harvest, or a backup after cold damage. If a fig is not fruiting well, fix winter injury, sunlight, and pruning first before assuming it needs a pollination partner.