Q:

How can I use the eOS terminal notification in Ubuntu?

elementaryOS has the following feature to notify the user of a finished terminal process.
Is it possible to use this feature in Ubuntu?

Figure 1: Example of eOS terminal notification

A:

What you want most probably already is available via your ~/.bashrc courtesy this and this:
Note: this assumes you have libnotify-bin installed. If it isn't, just run sudo apt-get install libnotify-bin to get it.
Check that your ~/.bashrc has lines like this:
# Add an "alert" alias for long running commands.  Use like so:
#   sleep 10; alert
alias alert='notify-send --urgency=low -i "$([ $? = 0 ] && echo terminal || echo error)" "Task finished" "$(history|tail -n1|sed -e '\''s/^\s*[0-9]\+\s*//;s/[;&|]\s*alert$//'\'')"'

Note that "Task finished" isn't present in the original code.
So, in your specific example, you'll run
sudo apt-get update; alert

If you want the notification to remain on-screen a little longer, use -t 3000 (in milliseconds). Or, if you want the notification to display until you dismiss it, use --urgency=critical.
Here's what I see: 

A:

undistract-me provides notifications for terminal commands that take longer than 10 seconds to complete.

sudo apt install undistract-me
echo 'source /etc/profile.d/undistract-me.sh' >> ~/.bashrc

