Any shade of blue between cobalt and navy. Indigo. Plum. Bronze. If you change the door hardware to oil rubbed bronze, turquoise.

I think the black door will look beautiful here. It will add character and match to another door and windows.

After you paint that door, remove those shrubs that don't add anything to the entrance and replace them with some color. Not sure where you are, but your local garden store can make good recommendations. Also, pots of seasonal flowers placed around the patio will also make your entrance a bright spot. Lastly, a piece of outdoor art would look good on that wall next to the window.

After you paint the door black, you might consider removing the shrubs which seem to be old and not thriving. I would suggest you visit your local garden store and see what is recommended for where you live and the space you have. I would also suggest that a piece of outdoor art would add some fun, hung on the wall next to the sliding doors. Lastly, some pots of flowers with lots of color would be a bright, welcoming addition.

I agree with the others, a dark charcoal or go BLACK! It will pop with your existing trim!