Hail Caesar?

This is a controversial subject, which is why only the historic aspect of it's origin is of interest here.

When the Nazis rose to power, they adopted the notorious stiff arm salute, which they actually ended up making mandatory for citizens in law. 

This is the exact opposite of the situation today, where doing it is illegal in Germany along with any promotion of Nazi-related activities or artifacts.

There are many people who believe the salute was widely used by the Romans, and its use by the Nazis was directly taken from this. In fact this turns out to be untrue, since although there are rare instances in Roman art and literature of something similar being used, it is not the case that it was widespread by them at the time.

Invented by the Nazis

The salute is shown in the early American movie industry whenever movies or plays were depicted in Roman times. It was carried on from an 1899 stage production of Ben-Hur, then on screen in the 1907 movie version.

Ironically, when the Nazi party did start to adopt the gesture, some questioned it believing it's Roman origins were "impure, since they were not of German nobility". 

The gesture was even used by other countries on occasions which warranted it at the time, for example the English football team during a game in 1938.

Hitler claimed that he personally saw through the adoption of the gesture and thought it appropriate for peace since it demonstrated the person making it was clearly unarmed.

The popular image of a Roman army making the gesture to Caesar by Hollywood is probably the biggest driver behind the notion that they really did this back in those times, and therefore the Nazis must have copied them.