Can I apply to a job my office mate got recruited for?
Why would you want to go after the job your office mate is interested in? It seems strange.
I think she's a bad person and I want her to fail in her new job.
It is fine to not like your office mate, but you should still be respectful towards them. Undermining their potential new job just out of spite is petty and immoral.
She is a bad person so it's alright.
If you have so much hate for her, why not just let it go so that you can move on with your life? Plus, her going to another job means that you won't see her face anymore, so why would you want to follow her to a new company? You are way too important and smart to waste your time on revenge.