A long kontos spear, used two-handed, was pretty much the norm for armoured shock cavalry in the Middle East for a long, long time. Often accompanied by a shield strapped to the left arm. It didn't give quite the same impact power as the couched lance, but on the other hand remained way more useful in the ensuing melee and eventually actually long outlived the one-handed couched-lance technique; light cavalry lancers were still employing a variation in the mid-1800s.

Such long two-handed spears were actually something of an international norm across Eurasia - even the Japanese started using them eventually.

'Course, a shorter one-handed spear worked right well too and was even more common.

The Muslims learned the tactical peculiarities of the "Frankish" massed couched-lance charge rather quickly, the same way the "Franks" living in the Outrémer were pretty quick to adapt to the peculiarities of Muslim battlefield tactics. The Byzantines were obviously no dumber. The thing was that neither adopted the technique, generally preferring a less all-out and more coordinated approach to shock cavalry tactics; the sheer shock power of the "Frankish" technique was however well respected, and whenever possible considerable effort went into eliminating it through the use of terrain, archery, light cavalry screens and whatever. Often, however, it was found necessary to simple meet it head on with friendly heavy cavalry; after they'd gotten over their initial surprise the Easterners proved quite capable of standing up to it if needed, although they preferred to avoid doing it if possible.

That's why the Ottomans were always so eager to employ Serbian knights.
The Ottomans were always eager to employ Serbian knights because those were, you know, in Serbia and thus readily available for combat duty on that theatre, unlike for example the elite kapikulu sipahi heavy cavalry that went with the Sultan and hence normally resided in the capital, or the Anatolian troops who had to be called up from Anatolia. Not that they were ever adverse to employing local troops anyway, especially on far-flung fronts. Remember that campaigning across the Balkans stretched even the formidable Ottoman organizational and logistical chutzpah to its limits - Vienna is actually closer to Paris than Istanbul (the main assembly and supply area of the Ottoman army), and that's not yet counting the difficult terrain of the Balkans along the way...

And I don't know what the Emperor's personal equipment has to do with standard cataphract equipment?
A lot, since he (and one would assume his bodyguard) wore a harness essentially identical to the standard cata equipement; full mail over which went the rather formidable klibanion (an improved Byzantine variation of the standard lamellar corselet), plus all the usual limb defenses and padding underneath.

That's more or less the standard equipement of elite armoured cavalry in most of Eurasia since Antiquity anyway, actually. Throw in horse bard. Muslim heavy elites employed similar schemes, and for example the Mongol elite heavies differed mainly in lacking the mail hauberk underneath and relying mainly on lamellar (often of leather; nomads tended to have trouble getting their mitts one enough iron).