The
Varangian Guard of the Byzantine emperors fall outside those dates (they were most prominent around roughly 1000-1200), but they were a military unit with a distinct reputation for skill, brutality and loyalty and were definitely considered elite. They were also an institution rather than a short-term warband. Edit: And, since this will become a recurring theme, they did support a usurper emperor at least once.
The
knightly orders of the various Crusades, like the Knights Templar, Knights Hospitaller and Teutonic Knights, were not only elite military units but independent or at least autonomous mini-states. Time-wise they are perfect for your European fantasy world.
The
mamluks of Egypt and other muslim rulers were slave soldiers, but also important members of the administration. In Egypt they eventually seized power, ruling the country from 1250 to 1517. They were not a single unit, but a soldier caste.
A similar system was set up in Turkey, with the
janissaries. These were the Sultan's guards (so more of a single unit, but also important parts of the palace administration), and eventually they too became kingmakers and power-players.
If you want Romans, the
Praetorian Guard of the emperors were probably considered elite (they were well paid and expected to be loyal). However, they did end up murdering emperors and choosing new ones...
The moral of history is that you really shouldn't have royal guards.