Please don't ask me for my password

No employee should ever ask for a password. The way is see it is that it's not a matter of corruption, or people eavesdropping on the connection (what a targeted attack that would be, targeting a specific, short, rarely occurring chat session). The point is that if your employees never require a password, you can put in big letters OUR EMPLOYEES WILL NEVER ASK FOR YOUR PASSWORD, and attackers imitating your employees are SOL, because they can't ask for the password without it being obvious that they're not real employees.

Surely you can just give relevant employees admin powers so they don't need the password.

If you don't trust them enough for that, it's relatively simple to set up a proof that the person has the right password without actually transferring the password itself, so that the employee can't change things about the account without express customer permission.

That's an absolutely ridiculous policy for them. No support person should EVER ask you for your password.

My employer asked me for my password once, for some site. I forget why - someone needed superadmin access, and this was the fastest way to give it to them.

I gave in, and promptly changed my passwords across all sites (this was back when I just had the Three Passwords) - and changed all of my work-related passwords to something stupidly simple, fighting off the urge to change them all to "fuckyou" and "shitslapbananamonkey".

This is bad, but not as bad as what I experienced with my hosting service---the employee I was chatting with straight up gave me my password.