Hi all! Can you help me diagnose my basil plant?

I just got one of those “grow your own” herb plants from the grocery so my expectations were pretty low, and it’s already done way better than expected.

I originally thought it was pollen from it flowering but I only had the one flower left after trimming and this stuff is everywhere. I’ve always grown basil outside and this time thought I’d try putting it in a window box (with cilantro and rosemary in case that matters) but I’ve not seen this before.

Is this something to worry about? If so what’s a good way to treat it?

More pics on Tumblr