I have also been researching this topic for a while. I was using Google hangouts for a while but lately it seems to be dropping messages. Tried Skype next but it will only keep message history on the cloud for 30 days. Next evaluated Facebook. It works quite well but I want to host my own on a VPS. The top three alternatives seem to be prosody, ejabberd and openfire. I was planning to use ejabberd but the issue I hav…
If you want to do this for real, what you end up doing is implementing your own server backend that receives XMPP messages and then sends out push notifications to a mobile client. Generally a pain in the rear, but do-able. Getting your own iOS developer key & dealing with app store approval for stuff like this is probably the hardest part (for shops that aren't already doing mobile development). There are 3rd-party services that help bridge this gap: examples are https://pushover.net/ or https://boxcar.io/
If you're okay with a hosted solution and don't really want to deal with development or deploying a XMPP solution yourself, many of our clients have had good success with Slack (https://slack.com/)