> That's not precisely true (...) Everyone here speaks Hochdeutsch too
Linguistically, Swiss speak Hochdeutsch even when they speak Swiss German(!) Which a lot of Germans of Germany don't when they speak their dialect.
So your argument is quite wrong and certainly not "precise."
The German speaking people all have local dialects and have to adapt when they move. But these are all "just" dialects. Ask Germans, they aren't surprised when they have to get accustomed. Look at the map here:
https://en.wikipedia.org/wiki/German_dialects
Note that all "upper" German regions are actually where "Hochdeutsch" (Upper German) comes from. Yes, "up" is on the South, down on the map. And the area includes Switzerland.
https://en.wikipedia.org/wiki/Swiss_German
"Swiss German (German: Schweizerdeutsch, Alemannic German: Schwyzerdütsch, Schwiizertüütsch, Schwizertitsch[note 1]) refers to any of the Alemannic dialects spoken in Switzerland and in some Alpine communities in Northern Italy." OK, let's check Alemannic:
https://en.wikipedia.org/wiki/Alemannic_German
"Alemannic is a group of dialects of the Upper German branch of the Germanic language family."
Finally, it follows that Swiss German dialects are actually all branches of "Hochdeutsch" family(!) They are just not the same as today's "standard German." Which is what people you communicate to probably mean when they say "Hochdeutsch":
https://en.wikipedia.org/wiki/Standard_German
See also DominikR's post here. He being an Austrian, his dialect also from the same "Hochdeutsch" family of dialects, wouldn't feel as an "immigrant" in Switzerland.