This thread explains regarding the fiance permit if you want to get married here:
Need help with marriage visa
Your spouse would get a one year B permit, renewable annually providing you stay married. Here’s some general info on marriage.
https://www.ch.ch/en/marriage/
They could also apply for facilitated naturalisation if certain conditions are met:
" I am a foreigner and married to a Swiss citizen. How can I apply for naturalisation?
You must have lived in a stable marriage with your spouse for at least three years, have lived in Switzerland for five years all in all, the last twelve months of which must be without interruption. If you live abroad, you must have lived in a stable marriage for at least six years and must have close connections with Switzerland. Regardless of where you live, you must be integrated in the Swiss way of life at least by analogy, comply with the Swiss rule of law, and you must not endanger Switzerland’s internal or external security."
https://www.sem.admin.ch/content/sem…06.html#a_0006
A word of warning if you marry an American or if you have a US Green Card yourself. Outside of the US you/spouse are still obliged to file US tax returns and could owe US tax on top of Swiss ones.
https://www.irs.gov/Individuals/Inte…-Aliens-Abroad
You/they will also have to sign a W-9 form for any “foreign”, i.e. outside of US, bank accounts you open and you could also find it difficult to get a mortgage here. And you/they would also have to file an FBAR form if any account/s come to more than the aggregate amount of $10,000 at any time of the year.
If you hold a Green Card it’s fairly easy to terminate using an I-407 form, but you would still need to file a final return for the partial year you held it and file an 8854 form to exit the US tax system cleanly. If you marry an American keep everything financial separate if you can.