My thoughts are simple:

Madara split in two. Though not in the way you all are thinking. Everyone here seems to be enamored with the thought that Madara can suddenly split like Muu. Muu's ability is his own, and probably not copy-able. Else, we'd have seen other badass ninjas do it but alas, we have not.

That said, I do believe Madara split in two. Though again, not in the way you're thinking. Stop thinking about the Muu frames.

Madara split in two because he used Izangi. By split in two, I mean, he lost one of his EMS sharingan forever. By losing one, he lost his EMS, MS, and probably most his chakra. Madara used Izangi in the fight against Hashirama. Hasirama had him beaten, as evidence by history. Though, according to Tobi this was a façade, the true goal was to simply get Hashirama’s DNA. What we don't see is a killing blow. The only frame shown was one of Madara with several swords stuck in him. If, hypothetically, Hashirama left the battlefield before Madara was completely dead he could have used Izangi to save his life, but sealing 1/2 of his sharingan forever.. (or so he thought..)

The one flaw is that to use Izangi requires senju and uchiha blood. Well, we already know Madara managed to "steal" his DNA during his fight with Hashirama, so there is no doubt in my mind he could have used the technique.

When we were introduced to Izangi, we learned several things. First, that Tobi knew exactly what it was. How? Well wouldn't it be easy to recognize a technique you once used yourself? Secondly, the technique itself, and it's power. If you believe the technique was "created" for Danzou to use and burn-out-and-die-in-a-blaze-of-glory you are sorely mistaken. It was introduced as foreshadowing. It foreshadowed the technique used by the person who acknowledged it's existence and knew it by NAME.
So who is Tobi? Tobi Is Madara. Tobi is the Madara that fought Hashirama, used Izangi to live, but lost most of his sharingan powers. He has extended his life by using Zetsu to become some-what immortal. Though it seems like it cost him even more of his humanity as he no longer bleeds.
As weird as it is to say this, but Tobi is Madara’s insurance policy. Who better to make sure you get resurrected with the rinne tensei than yourself? I have no idea what Ressurected-Madara’s plans were for Tobi upon resurrection. Perhaps the Rinnegan (or the EMS from the resurrected Madara) have a power to combine souls?

Why was Tobi horrified when kabuto showed him Edo-Madara? Tobi was horrified because he was supposed to be given a true resurrection and not an Edo one. It was not because he was afraid that people would find out he wasn’t Madara. He IS Madara, he was upset because he did not want his one chance at regaining his “True Power” lost forever because Kabuto wanted to toy with him.

Edo Tensei (And, Rinne Tensei) bring back the user to their full original status. Meaning, dying and being Tensei'd was the only way for Madara to become "Whole," retreiving his blind Sharingan and restoring his EMS along with his Hashirama DNA making him the Sage of the Six Paths Reborn. Although, there is a difference here - a VERY VERY VERY important difference between the two resurrection techniques. Tobi AND Edo-Madara have BOTH stated that Rinne-Tensei ONLY was to be used ON or FOR him. You could argue, that if both would have restored his EMS, why didn't Madara just become Edo-Tensei'd back sooner? Well, he does not want to be controlled. Simple.

But how can you edo-tensei someone who isn't dead? I think Madara was a smart man. I believe (as someone here has already thought up as well) that Madara sealed his soul within his eyes. When he used Izangi, half his soul was "lost." Thus, allowing him to live, but also to later be resurrected and restored. Perhaps achieving EMS fuses your soul into your eyes as a by-product.

This post very much explains why Tobi refers to himself as Madara in the strange manner that he does. He is, and he isn’t. Rather, he is no longer what he was, and therefore, is

As Itachi Uchiha put it,
“A shell of his former self.”