This was one of the main concerns about moving to Slack in the first place, so I understand the question. However it turns out there are a couple of ways to handle this that I think are pretty simple and relatively unobtrusive.
One way is to download the Slack app for your phone, if you have a smartphone, and set it up to give you notifications. That way you’ll be pinged if something gets posted (you can adjust what events cause a notification, and shut off notifications if needed).
Another way is that you can configure Slack to send you email notifications of direct messages or mentions. Here’s the official page with instructions: https://get.slack.help/hc/en-us/articles/201649273-Email-notifications Note that this does NOT send you notifications just of any posting whatsoever, just if someone messages you or mentions you directly. Those would be the notifications that typically you would want to respond to the most quickly. Then, for regular postings, if it were me I would set a recurring reminder on my computer or phone to remind me once a day (or however often you prefer) to check in to Slack. In this way it’s like an email thread — the posts are there to be read in the same place when you have the time.
I’m basically busy all day as well and my solution is to leave the app running on my computer in the background, so I get an alert if someone posts; and when it’s time to go into lockdown mode and I want no interruptions, I quit the app but set a reminder to myself to check in later, or just do it the next day. There is a little horizontal red line in every Slack channel that shows where you last read a post, so it’s fairly easy to scroll back and pick up where you left off.
I personally don’t necessarily want to know when ANYTHING happens on Slack, just that I need to remember to “check my email” as it were on Slack at regular intervals, and setting up a recurring reminder on the computer or my phone seems like the easiest way to do that. If you want to know more about how to do that, please let me know.