Multiple books that cover the same time period but with different characters can work and it has been done, the problems are:
1) Each character will have a different voice and be a different personality (or at least they should be). As a result if you re-do the same time period from 4 different perspectives you can end up with a character readers "hate and characters readers "like". Now in a series where you've multiple points of view through the book the characters readers hate are gotten through because they know a character they like will be coming up soon. Indeed in a good book its more the case of like and love rather than like and hate outright.*
2) Continuity is an issue, but so long as you flesh out the master plan before you write it shouldn't be a problem. Each character can have minor events that shape them as a character, but which play no part directly in the greater scheme. If you fail to create such a master approach then your writing will show it. The first character will get all the action whilst each one there after will have increasingly less and less choice and might well up in plot holes and problems that are just going to force you to do quick-fixes that don't sit right.
3) Writing voice - readers like to pick up a book in a series and have the same "voice" through the whole series. A risk with doing 4 different angles is that your writing style might change to suit the character; this might throw some readers if it happens in a very clear manner (again this supports the notion of writing all at once and then splitting - or at least having the bulk written and planned out in rough).
4) Missing complexity - Complex interactions are great, but if you spread them out over multiple books then the average reader is only going to remember a few key events clearly in their mind (esp if its been a while since publication of the first to second book). As such you might put lots of subtle key links between them that simple get glossed over on a first read through. Because most readers don't want to have to go back to the previous book(s) to keep up.
Doing multiple points of view in the same book you can do the same effect, but without the same forgetful aspect on the part of the reader (and checking is just flicking through a few pages).
5) Pace - if you do four books all covering the same time period then pace within the story can be an issue because you can only cover the rough same time period over and over. This means each character has to have their own final ending - their own conclusion.
I have seen it done where each character splits off in their own tale within the world - each reaching a conclusion of their own that leaves them "ready" for a final book that combined the characters; drawing them together to wrap up the story.
6) Depth - if you do a book per character you have a lot more room to go into deeper depths of the character. You can flesh them out a lot more because you've a lot more room to write. If you do all in one you can still take that time, however you might find that you can't take each one too far apart from the others to go into their own personal background without losing the cohesion**
In short a great part of this is how you want to write it. There is no correct nor right way to write it (unless you ask a publisher who will tell you what they want to see based upon their focus and market research) thus its down to you. Get a plan together; get segments in draft and see how you feel about it; see what will fit your ideal structure for the story.
*Note like/hate/love etc.. I mean in descriptive terms - sometimes the real evil nasty character is the one lots of readers like to reader about
** If you think about books like Game of Thrones whilst multiple characters are often interacting far apart from each other; they do keep meeting up. Each "area" has a collection of characters interacting not just one alone for the most part. As a result you never get the feeling that one character is "off on their own". You can do that for a time, but do it for too long and hte reader will start to wonder why this character, so far removed, is doing nothing much with the main story.