How to do a 45 degree corner on a privacy fence?
I don't see any need for a second post. Here's how I would do it. Incidentally, buy some 3" deck screws for this if you don't have any. You will WANT to use screws here, even if you've been nailing


Set the post square with the bottom part of the fence in your diagram, but extend the 2x4s all the way across to the edge of the 4x4
Take a miter saw and set it at 45 degrees. Cut one end of your next boards vertically so that the saw goes through the corner of the board (right hand cut with the board standing up)
Dig the hole for the next post (along the angle). Put the 4x4 in but do NOT set concrete yet. This hole may be slightly further apart than the previous ones, so take that into account. You might want to dry fit the 2x4s first just to make sure (re-digging holes is a pain)
Have someone hold the 2x4 you just cut on the end by the 4x4 you just added. Screw the beveled end into the 2x4 you ran across the previous 4x4. Be careful not to get too close to the edge of the beveled board or it may split.
With your compatriot still holding the 2x4, get the 4x4 level and then attach the 2x4 to the 4x4 (leaving room for the next set of boards). Repeat for however many stringers you're running
Pour concrete for the new 4x4 (your stringers will hold it in place) and now you're ready to run your 45 degree side from there.


Congratulations! You've just made a 135 degree turn



Now, if you're scared about the 2x4 splitting (or you got the screw too close to the edge), what you can do is take a spare piece of 2x4, cut some pieces (same number as stringers) the same size as the 4x4 (about 3.5") and bevel it 45 degrees the other way (left hand cut) before you start step #4. Screw these onto the 4x4 and then you can sink 2 additional screws perpendicular to the board you're placing, though these additional pieces and into the 4x4 itself. This will guarantee you a strong corner, but it can be overkill. I'll leave it to you if you want to do this extra step.