Almost everything functions on my page www.neesconsult.de/ .
If i select english, the english page is shown.
The language is kept, when I select another menu or content. Only when I select the "home"-page, the german page comes back. This occures, when I change the language on a page different from the home-page.

What can I do?

Could you please send a screenshot of your JoomFish related settings in JoomSEF and also tell if JfRouter plugin is active or not?

I assue home for german is the main domain, what is the home of the english at your site? Is it /en?
If so, then it is logical that the language switches to DE when visiting home as each page in each language should have just 1 URL. So you cannot use the one URL - just the domain name both for English and German.

Yes the home for german is the main domain, and the home of the english at my site is it /en.

But what can I change? I do not want to have two domains,

Hello,
well I am not sure how you go to "home" page, but mostly it is enough to adjust the template code so that when you have DE active and you click home, you will go to main domain / and when you have EN active, you will go to /en.

This however needs to be ensured by correctly created template, which means you cannot using simply "/" as the link target but you need to operate with Joomla's site base variable.

in the configuration my homepage is neesconsult.de.
But what should I change in the template to get the right behavior?

You could use code similar to this,
1. first, get the default menu item

2. use the default menu value