You might want to look at Fastmail.com, a paid e-mail service with actual customer support. They provide a web-based interface, mobile apps and also IMAP access, and you can use your own domain with it if you like. (I've been happily using it for a few years now, and first heard about it here on HN.)
Fastmail is too expensive. They have no competition. I wish someone would come around and disrupt those prices.
While their offerings differ in the details, by and large they offer the same thigns as fastmail.