If you liked Mailbox's inbox zero concept, as well as its customizable and varied quick swiping inputs, your best option is Spark by Readdle.
Most notably, there is no way to change the Outlook icon badge to display the number of all unarchived emails.
The Outlook app badge count will either display the total number of unread emails, or it will show unread emails only within Microsoft's "Focused Inbox." The Focused Inbox feature intelligently interprets which emails are personal and important, and which ones are likely just mass emailings of less urgency.
Google itself offers a pair of email clients that are both robust and functional.
The official Gmail app delivers what you'd expect, including support for multiple accounts, as well as instant push notification alerts for new messages.
Some third-party i OS apps address this issue by offering instant push alerts from emails.
This is accomplished by having constant access to your Gmail account on their servers.Google Apps is priced starting at per user, per month.Of course, there are many other i OS email clients with push Gmail support, some of which are also popular and well received.The simplest way would be to have your Gmail forward to an i Cloud account, though there could be delays associated with this. After receiving the alert, you could then open the Apple Mail app to read the note.There's also a service called Nueva Sync which adds push notification abilities for Gmail with Apple's Mail client. Finally, a Google Apps subscription adds Exchange Active Sync support.Swiping from the left (or right) side of the screen flips between emails rather than going back to the previous menu, which we do find to be annoying.