Short polling in the background + notifications for DMs incoming from private / auth enabled relays. Other than not trying to write all of my notes to the inbox and DM relays, as mentioned by nostr:nprofile1qqsw9n8heusyq0el9f99tveg7r0rhcu9tznatuekxt764m78ymqu36cpzdmhxue69uhhwmm59e6hg7r09ehkuef0qy2hwumn8ghj7enjv4h8xtn4w3ux7tn0dejj7qg4waehxw309aex2mrp0yh82arcduhx7mn99ueat4s3, this is the single quality-of-life improvement still missing for me.

As for longer-term features, I would vote for having a build with an embedded, pre-configured "native" Android relay, à la Citrine, exposed to the Internet with a public reachable onion address and pre-configured with sensible filters so that every Amethyst client also becomes a self-sufficient personal relay out of the box. This would be the dream.