You could consider Obertauern or Lech am Alberg in Austria. Like Mürren, both sit at a relatively high altitude. Lech is in a high valley, Obertauern sits in a saddle at the summit of a mountain pass, higher than both Lech and Mürren.
Obertauern doesn't have the cozy feeling of Mürren, because it was purposefully built as a ski resort. It is quite pleasant, though, and the town advertises a mountain view in every hotel room. Many of the hotels close during parts of the summer, but those that remain open offer exceptional value for their quality, including outstanding full and half board meals. The town can feel kind of empty in the summer, but I found it refreshing to share the mountainsides with more livestock than people. Don't expect any nightlife. Also, if a heat wave strikes during your visit, you can all but guarantee that temperatures will remain comfortable here. Conversely, on a cooler than normal summer day, temperatures can drop pretty low. You would need a rental car to reach the town.
Although not as compact as Mürren, Lech has a much more traditional Alpine feel than Obertauern. This being Austria, expect some excellent full and half board meal options. It's not completely dead in the summer, but this lovely little town seems to have escaped the notice of most of the tour bus and Ricknik crowds. It's also easier to reach via public transportation than Obertauern. Lech also has a satellite village called Zürs further up the mountain pass, although I think the hotels may only open for the ski season.
I spent a few hours in Passo del Tonale in the Ortler Alps of northern Italy. Once again, a modern ski resort, amazing view of the peaks if not that much charm in the town itself, and high enough of an elevation to escape even the sweltering heat of Italy in August. One of the more interesting war memorial I have ever seen sits high above the town on a ski field. You can see remains of what was the front lines between Italy and Austria-Hungary in WWI. A tunnel that served as a bunker during the war now houses a rather terrifying little museum highlighting some of the typical sounds of the war.
The adjacent Val di Sole has several nice little towns, the most Mürren-like of which is probably Peio.