Is it possible to build an amp site without the need for a complimentary "normal" site?
Yes you can build a standalone site in amp. You can have example of AMPs own website where desktop and mobile versions are built in AMP only (https://amp.dev/).
Since often people having site are moving on to build a AMP version of site you may be having this question. I would suggest if you are planning to go for a new website where user experience is more important like blogging, news or eCommerce, Go for AMP. I am not promoting but realized it when build for myself.
Mobile version of site can build with AMP + PWA and it will give you more a mobile app kind of performance.
Validations of amp components are little strict but they also make your work less, No need of writing complex JS and designs too.