What would a "Perfect" inductor be like
The perfect inductor has reactance without any resistance. In other words, the real component of its impedance would be zero. Loss of power as heat within the inductor is thus also zero.

The perfect inductor presents no impedance to a constant current (i.e. DC), yet opposes any slightest change of current. Any non-superconducting material can not meet this condition, as it is bound to have some resistance.

Hence, a perfect inductor would need to be made of superconducting material.

Advantages at the trivial level would be elimination of any wasted power through resistive heating in an inductor, of course. Beyond that, one enters the realm of speculation: There may be many advantages, but also design challenges.