Fractional order system is widely familiar title in FC

Technically speaking, what is the most suitable in your opinion?

Fractional order system is the most used notation by the scientist community, the non-integer order system may be it was firstly introduced by Oustaloup "CRONE : Commande Robuste d'Ordre Non Entier". For me that is the first time when I hear generalized system to signify a fractional-order system.

The word "fractional" is a misnomer because it could mean as either a fractional value between 0 to 1, and any arbitrary value greater than 1.

I think the best answer is "arbitrary-order systems"

When talking about arbitrary order systems, it can be a complex number no?

Yes it can also be a complex number. I remember reading a paper a few years ago where the order is a complex number, e.g. 3.2+j5.6