well, it depends on how much weirdness you are happy to put up with in a base :)
The problem with non-integers is that either there are numbers you can't represent (even irrationally) or there are multiple ways of expressing the same number (I guess this is actually true of all bases: 0.999...)
Base phi is the second; it happens in quite an orderly way, so it's fairly usable.
If you look up #basephithememe I've got more of an explanation of how it works up there somewhere
A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes.
\) for inline LaTeX, and
\] for display mode.