The game Suspended was a game in which you controlled 5 different robots, but each robot uniquely had one sense - one could see, one could hear, one could touch, etc - and you had to coordinate them to solve puzzles requiring many senses.
In that way, this game features multiple characters with unique skills, chief among them is a unique way to interpret the world. This justifies the split-screen aspect by actually giving you unique information in each screen. In this example, the robot Gizmo has the ability to detect electrical pulses, tracing the wires in the walls. Bruce can't see these, so having Gizmo tag along gives the duo the benefit of seeing the world in two different ways simultaneously.
Other visual features might include a blind person who can only see sound (much like [#085 - Sonar Battle]), or heat vision (allowing players to detect androids posing as humans), or perhaps even night vision, to lead other characters through the dark. Even without special vision rules, interesting puzzles can still be set up by having different characters seeing different parts of the same event simultaneously. For instance, to break a character out of jail, you need to see when the guard falls asleep from inside the jail, have a character on watch outside the jail, a character using security cameras to keep an eye on the people in the building, and a character to actually navigate through the structure and do the rescuing.
Probably the greatest problem with four split screens at the same time is that it would either require significant micromanagement to coordinate them or the opposite issue, having three characters stare into space while you concentrate on just a single one. To combat this, the characters in the game would be autonomous to some degree, making the game somewhat more like a fishbowl sim when characters are idle.
Essentially, you can give goals to the characters that they will carry out without you having to control them every step of the way. For instance, you can bring up a map and tell a character to go to a specific room and they will navigate their way their without you needing to click on every door. You can have them follow another character around and use their screen for a secondary pair of eyes. You can tell them to ask around about a specific topic and they'll wander around talking to characters until they find out some new information. If they become idle, perhaps they'll just wander around and interact with the environment.
Of course, if you ever need these characters, you can instantly take command of them. But when you aren't using them, they become autonomous. Perhaps they even have personalities where one idle character go shopping while another will locate the nearest pub and start drinking and gambling. Even if they aren't actively working towards a specific victory goal, they should still exhibit their own personality as well as bring out the personality of the world when idle. Having an idle character walking around flirting with girls and getting slapped brings nothing to the gameplay, but defines the character and world while simply being amusing to watch as well.