Your browsing history is held by your browser and crosses the various connections that were used. If you browsed site X on your work computer at work, then later were connected to another network (which one does not matter), site X is still in your browsing history. If you browsed site Y while on that other network, when you later connect to the work network site Y still remains a part of your browser's browsing history. [All of this presumes you are not using private/incognito mode, which does not store browsing history at all, nor cookies, after the browsing session is closed.]
Generally any company owned equipment is subject to at-will inspection, either by humans or software, by the company. This is entirely reasonable since company equipment is rightly expected to be used for company business. Most companies are not stupid enough to nail anyone for an occasional "boredom browsing" or "while I'm waiting for . . . to complete browsing" that is not excessive.
If you are using incognito mode there should not be much that can be easily traced by "easy" forensic methods (as opposed to law enforcement ones) provided you are not connected to the work network. Anything on a given network could be (and often is) monitored by the entity that owns that particular LAN, and that includes if you're in incognito mode since traffic still has to go over the network.
And all of the above is predicated on the company not having some kind of logging software installed that tracks what has occurred as far as web access at the machine level. If it does this for any and all browsing then all bets are off as far as having anything be in the slightest bit private. Such software does exist.
If you wish to avoid these issues you really should limit the use of company issued equipment to company business, regardless of what network you might happen to be connected to at the moment.
Edited by britechguy, 07 March 2018 - 03:11 PM.