A high alt version that was designed to use similar parts to the 109 while getting the same high-alt performance of late-war planes that were delayed or running into problems.
It was a 1943 design based on the 109G but with pressurized cockpit and much longer, redesigned wings. Initially a protoype test showed some promise but then subsequent versions showed less and less promise. Eventually some pre-production models were tested with 1-off assignments here or there sometimes in the recon role. However, it was deemed unsatisfactory and you have to remember at the time that things like the Fw190C and much better high-alt designs were already being tested, things like the 109G-5 and other designs were in the works with more horsepower and better performance. The 109Hs were built around the 109F engine, the DB601E, and some versions were designed to use the Jumo, but then along came the DB605 and later models of the 109G got a lot better.
In short it was a never-was design. It wasn't a could-have-been. They made a few, it flew but not as a frontline combat plane, or even as a fighter from what I read. It was outdated when it was built and it never progressed past that point so they just used the few airframes they had because a war was underway and it (literally) met the minimum requirement: It could fly.
