PHPStorm plays nice when any framework. The issue it's going to have is when Frameworks do not clearly document, or when they make bad decisions about things. You can, of course turn off different levels of inspections to account for the project you are working on. However, I've found that for the most part, if PHPStorm is flagging something like "undefined method," it's essentially saying "you are hoping that this works as expected." It's a form of static analysis.

I'm not surprised that an old CodeIgniter project flagged these issues.

As for vi, I use vi every day as well. I switch between that and PHPStorm. It depends on the context that I'm in. Both are wonderful tools. But I can assure you, auto-complete is a basic feature of PHPStorm. It's akin to 'set nu'.

Edit: This comment comes off as a bit snarky. It's not intended to be. I think JetBrains and it's line of products are excellent, and I've only ever had good experiences with them. I wouldn't want my comment to reflect poorly on them.