Lagerstroemia indica, Crepe myrtle Purple, Fast growing shrub or tree, 30 fresh seeds November 24

Germination method
Crape myrtle seeds will germinate most any time, but they will do best in early spring when the days are lengthening. Gently press them into the surface of a light, moist potting medium. Cover with a layer of milled sphagnum moss (or the same medium if you don't have it), and mist until damp. Cover the pot with plastic, and put in a warm, bright place at 20-25°C.

You can also try baggy germination method. Spread your seeds between two paper towels. Place the paper towels in a plastic bag or container (I upcycle a lot of containers from sweets and nuts, well washed with dish soap or bleach before use), then wet the towels with water using a spray bottle (adding 1 part of hydrogen peroxide to 4 parts of water will help to avoid fungus). Then seal the bag or container and keep at 20-25°C. Check regularly until tiny roots appear and transfer into pots or seed tray using plastic tweezers.

The seeds should sprout in a two to three weeks. Once they sprout, you can remove the plastic and keep the seedlings moist and in bright light as they grow. Wait until they have two sets of true leaves before transplanting to individual pots. Keep the pots indoors until late spring, then move them to a shady spot outdoors.

There's different recommendations regarding the stratification in order to obtain better germination rate. Some do it without stratification, others suggest 2 month of cold period.
Well I can tell from my own experience that the seeds that were stratified for 14 days had higher germination rate (close to 90%) comparing to those sowed without pre-treatment.
You can always try different methods and share them with me, I'll greatly appreciate it!
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Happy gardening!