Nope, you can run with Test Complete, and would only need the mobile license if you're testing mobile applications from it.
Hi,
The generic requirement for all applications that have UI and interact with end-user is to have an active desktop session.
To fit this requirement (and note that it is not TestComplete-specific, but is valid for Selenium and any other GUI application):
a) There must be active, not locked desktop session on the test computer; and
b) Jenkins agent must be started not as a service (because there is no active user session in this case even if you allow service to interact with the desktop in its settings), but as a regular application within this session.
See https://support.smartbear.com/testcomplete/docs/testing-with/running/via-rdp/index.html and https://support.smartbear.com/testcomplete/docs/testing-with/running/on-locked-computers.html for more details.
P.S. Also, Smartbear provides Jenkins plug-in for TestComplete - https://support.smartbear.com/downloads/testcomplete/jenkins-plugin/ (https://support.smartbear.com/downloads/testcomplete/)
