Strictly theoretically, you could do any action. It's just that if some of your ideas are not implemented already you'd have to wirte a custom alert action.
But.
You have to remember that with Splunk you only connect with your browser to the Splunk environment. That's one thing. And alerts (not only in ITSI but in Splunk in genral) are simply generated by Splunk "internally" by running a scheduled search and acting on its results. So for alert to be generated there doesn't even have to be anyone logged in. And if there are several people logged in, who should be alerted?
It's not that easy because Splunk has several components and the one directly facing the user (the browser with Splunk GUI) is not the one doing "the work".
