I'm not sure what you're looking for.
For me AquaMail runs like a charm:
Scheduled sync every 15 minutes
IMAP Push delivers and notifies me immediately about every new incoming message - even between scheduled sync events -
5 accounts - all configured as IMAP
1 Gmail account - also configured as an IMAP account
I'm pretty sure that your issue is caused by some weird configuration / settings.
I recommend to uninstall AquaMail and made a clean installation / account configuration. Then please have a look into the app's FAQ-Site at:
http://www.aqua-mail.com/?page_id=227and go through the following items (search for "..."):
• How do I enable IMAP Push
• Notification icon in IMAP Push mode
• Notifications about all unread vs. just new messages
• Can I exclude a particular account from scheduled mail checks, without having to delete it?
• What does the 'System settings' setting do?
• How can I disable scheduled mail checking in the background?
• How can I specify my own mail check interval
In addition, you will find many other useful information on particular subjects and situations in the FAQ.