I am an employee of a hospital in Maryland, but recently moved to California and was granted permission to work totally remotely for them. Will this affect my paycheck? Also, how does it affect my tax return process?
Your income is subject to tax by the state that you work in, and by the state that you live in, if different. But in most cases it doesn't matter what state your employer is in. What matters is where you work and where you live.
You no longer work or live in Maryland. You now work and live in California. The money that you earn for working in California is taxed only by California, even though your employer is in Maryland. They should no longer withhold Maryland tax from your pay. They should withhold California tax. If the hospital will not withhold California tax, you should make estimated tax payments to California so that you don't owe a huge amount of tax, plus penalties, when you file your tax return.
For the year that you moved, you will have to file part-year resident tax returns for both states. For subsequent years, you will file only a California resident tax return.