I use the mailbox iPhone app. The killer feature for me is that I can snooze an email. It's close to ideal for me. This ability to show multiple inboxes on the desktop might be an interesting tool to add visibility to something on another list.
Can't beat Mailbox. Wish they had a desktop app too. The snoozing thing is essential for moving stuff out of my headspace until I can think about it. I wish everything in life had a swipe to snooze.
I enjoy using Mailbox to the point that I rarely use desktop Gmail anymore -- a keyboard on my iPad helps. It's also easy enough to implement the filtered inbox views mentioned in the article for Mailbox folders, allowing the same sorting functionality on mobile (unlike the star-based solution).
I recently read that a desktop app is in the works. I'm sure they are working like crazy on it.