Greeting all,
I've been having a similar problem but my symptoms have been: open iTunes, dock icon bounces but a window doesn't open. I discovered that iTunes was actually running by hitting the 'play' button on my keyboard.
So, no exactly everyone else's problem, but still a major annoyances. I tried a little of everything: re-installed 12.3, re-installed an older version, permission repairs, shutting off/re-starting iCloud sync, Nothing worked. I installed the El Capitan 10.11.1 and iTunes 12.3.1 on Wednesday and again tried to open iTunes and still had the same problem. I left iTunes running (?) and this morning the iTunes window was open and working.
I'm going to speculate that the 12.3.1 update had a fix but had to re-build the library, or something, before began working.
Install the update, open iTunes and give it some time before you try something else.
Good luck everyone.