Per the EDD, businesses that employ one or more employees must register with them if they pay more than $100 in California wages per quarter. Depending on your employers business type, requirements on registering with the state may differ. Please click here for more information.
Once you become a California resident, the wages you earn working for your D.C. based employer (as well as all of your other income) will be taxed by your new resident state. Your employer must withhold California state income taxes from your pay.