Washington D.C. and Washington state both took their name from George Washington. How much their flag should look like? Firstly, look at District of Columbia flag:
[Current flag - Design by Charles A.R. Dunn]
This flag I won't change. It's represents the DC incredibly well, and has a very thought origin: it's a rectangular version of the coat of arms (i.e. a banner of arms) of the family of George Washington, that gives the federal district the name.
Now, let's see Washington flag:
Like DC flag, Washington state flag pays tribute to the USA independence hero George Washington, but in a very lazier way. As nobody wants Washington's face badly conceived on a flag, I think we should work it out. The easiest way to do it is mix Washington arms with state colors, like in these flag by grendelsbayne (very similar to one by Andrew Rogers):
[Proposed flag - Design by grendelsbayne]
This design is outstanding, and could do an excellent Washington flag, but I think I can do something by myself.
[My proposed flag]
A new arrangement, remarkable but still related to Washington arms.
Your comment is welcome!