After the split of the Roman Empire in 285, did people still look at the west and east as one big empire, politically split into two countries or did they see it as two separate large empires as we do today?

Contemporaries definitely viewed the two halves as two parts of one empire! The division in 285 wasn't really meant to be a proper split, but a way to distribute the responsibility of the empire onto several subordinates - Diocletian was still the paramount ruler, as he was for example able to force the resignation of Maximian and made the other tetrachs follow his plan for succession. One only have to look at the propaganda released by the tetrachs (which constantly proclaimed how united they were for instance), even their names (all of them adopted parts of Diocletian's name), to get a sense of how united they were, at least in theory. Things got more complicated after Diocletian resigned, but various emperors continued to co-operate in terms of legislation and strategy when it suited them, especially when they've come to an agreement to attack their mutual enemy. One famous example would be the Edict of Milan, which was the work of both Constantine and Licinius, and applicable to the entire empire once Licinius got back to his half and ratified it. Constantine later reunited the two administrative halves of the empire, likewise for his son Constantius II, so personal rivalries between emperors never really solidified into a firm political split.

After the brief reigns of Julian and Jovian (both of whom ruled a united empire), the Valentinian and Theodosian dynasties dominated the empire, so the empire really was still one entity despite occasional bouts of infighting between family members. Laws for example continued to be issued in the name of all the reigning emperors (even though in reality they never applied to the entire empire). As we can only really get at the mindset of the elite, the fact that the imperial bureaucracy kept pushing this idea of imperial unity tells a lot about what the aristocrats thought about the empire.

This got a lot more complicated in the mid-fifth century, but to what extent can we attribute this increasing divide between the two empires to different views of the empire? Imperial failures in the west made it weaker, which was very different to the rich and thriving east, so it definitely looked different and led to different responses, but we can't really say that the elite thought different - propaganda continued to trumpet unity, even though in reality things rapidly deteriorated. Furthermore, just take a look at the actions of various Germanic rulers post-476, as they all acknowledged the emperor in Constantinople, even though their governments were fundamentally based on western traditions and guarded western interests. So yeah, I don't think there's any reason for us or contemporary writers to see them as two distinct entities - the Roman Empire was literally all everyone knew, so to a certain extent people weren't even able to conceive of a world without its overarching influence!