We have something similar supported called IMAP-IDLE. IMAP-IDLE allows a client to indicate to the IMAP server that it is ready to accept real-time notifications. IMAP + IMAP IDLE allows for immediate email
notification and delivery service of email. Very much like push, it's low bandwidth, good on battery life, as well as being widely deployed. However, at this time, I understand the iOS default mail client does not have support for it. Android does, among a large list of clients for your computer.
The Wikipedia page for IMAP-IDLE lists a number of clients and more information on how it works.