Heat pump on a pedestal or directly on the floor?

We offered them on an aluminum base just like you did. The reason is probably decoupling. Relatively unspectacular but it works.

And is something moving there? Do you maybe have a picture of what something like that looks like then?

With us, there is also a layer of rubber between uncoupled concrete slabs.

I would always decouple devices with the potential to become structure-borne sound sources, because I would not want a rumbling throughout the entire house. Better safe than sorry.

Yep, I would always decouple. A concrete pedestal is not an incredible expense. Everything else is, in my opinion, saved in the wrong place.

I also heard you can put a mat underneath, like with a washing machine. My neighbor has it like that, or rather it was done that way by the tradespeople.