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For Webmail or IMAP mail, look no further than Fastmail. These guys know exactly what they are doing with IMAP and they can easily accomodate your multiple accounts. I've been a Fastmail user for almost a decade. I've never been happier with any email provider (or any provider, to be honest). You pay for the service, but it's worth the money and more. You are treated like a valued customer and the support, if needed,…
How does it compare to, say, Gmail enterprise? Well, apart from the whole Google Apps. I'm curious because we'll have to migrate to a better email provider in a month or two, and Gmail enterprise seemed to be the best fit for us (price, storage, good iOS app to leverage push notifications, etc).
On push, Google Apps gives you ActiveSync as a protocol, as an alternative to IMAP. But the integration with iOS Mail is weird and I could not use it.
But FastMail on the other hand, in a twisted turn of events supports push through Apple's own APN, getting the same treatment as iCloud: https://blog.fastmail.com/2015/07/17/push-email-now-availabl...
FastMail also has a native app that does push notifications, but it's just the packaged web interacts with push notifications added. This is both good and bad. It's good because the mobile web interface is very decent, compared to Gmail, and this means you always have a decent UI on whatever OS you have. It's bad because it doesn't feel native, but then there's nothing more native than the iOS Mail app.