is there a complete test command script that allows you to duplicate the current session in a new browser tab?
TestComplete, as a UI Automation Tool, has the ability of sending script code at web pages and such... but generally, as @BenoitB mentions, your easiest bet is to simply do what a user would do... send the appropriate keystroke to duplicate the tab in the browser.
Robert Martin [Hall of Fame] Please consider giving a Kudo if I write good stuff ----
Why automate? I do automated testing because there's only so much a human being can do and remain healthy. Sleep is a requirement. So, while people sleep, automation that I create does what I've described above in order to make sure that nothing gets past the final defense of the testing group. I love good food, good books, good friends, and good fun.