Re: Building a covered patio, how deep should posts be?
As a fellow Californian and a semi-retired contractor, I offer this:
DO NOT set the posts in concrete, rather, wet set a steel, elevated post bases into a pool of concrete that is at least 12" square and 12" deep. If your structure increases in size, then increase the depth accordingly. The reason for this is that wood set in concrete is going to rot, PERIOD, whether you use a pressure treated post or not.
From there, use at least a 4x4 post. According to your plan of an 8' square structure a 4x4 will suffice, much larger than that and you'll need a slightly larger post.
Because you've used a pier style support base, you'll need to build structural stability into the arbor/patio cover and that is done by using knee braces (angle braces ) at the top of the structure where the post meets the horizontal "cover".
I would recommend that you Google "patio cover", "arbor", and "pergola" for ideas and designs. While the structure you are interested in is not difficult, it is important to incorporate both design and structural components to achieve a sound design and aesthetic result.
Please post back with more details so that we may further assist you with the design and implementation of you project.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!