Gut check: tell my "cool boss" I'm looking for other work?
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I think most people will advise you not to tell your boss and that is definitely the safe approach. I’ve had a similar relationship with my first boss/mentor and decided to let them know, but only after I had interviewed and was close to receiving an offer. On 2 different occasions, they actually talked me out of taking new jobs and gave very insightful guidance about the trajectory of those opportunities and why I should think twice about them. I was disgruntled in my current role and obviously suspicious of their bias to keep me around, but looking back it was the best possible guidance I could have received. Later, when I finally got an offer that I was excited about, they validated my excitement and encouraged me to take it. Taking it a step further, when they left that company and went to a new company years later, they recruited me to join and we're back working together.
Point is, if you’re fortunate enough to have the kind of relationship that weights more heavily towards mentor/mentee vs. boss/underling, you should take advantage of that and look to preserve it. Only you will be able to make that judgement but I can't overstate how important having a really strong relationship with a mentor has been for me in my career.
Everybody here will probably tell you not to inform her and emphasize the downside of potentially pissing her off. I think the optimal approach here depends on how intimate your relationship with this person really is.
I'd encourage you to reflect on whether this person is someone you truly consider a 'mentor', or if it's just someone senior who you have a good relationship with - that's a really important distinction and i think too many people confuse the two.
If they're really a mentor, your best approach is to sit down with them and discuss why you're thinking about exploring other options, framing this as more of a 'i'd love to get your feedback and advice' type discussion and less of a 'I don't think i like it here anymore so i'm trying to move on' declaration. Worst case scenario she's aware you want to leave (which you'll have to bring up at some point either way), best case scenario you get extra respect from her for being transparent and trying to help out the team (which can go a huge way over the longer term of your career).
70-80 hour weeks for what sounds pretty much like a standard divisional FP&A role is certainly not standard. This is a corporate environment, not IB. If "big boss finance lady" were such a big boss and a good manager, she would make sure that her team is appropriately resourced. It seems like she is either disinterested in the WLB of her team, or lacks organizational clout.
Recruiting taking 4 months for a junior analyst role in the current job market? Doesn't sound reasonable. And sleeping in the office? At $83k???
Bottom line: You owe absolutely nothing to her.