Muu's Fission Technique
We all know about Muu's Fission Technique: This technique allows user to literally split his own body into two identical yet entirely independent duplicates of himself. Unlike Clone Techniques that only appear to be similar, the ability is instead a physical separation and so does not require hand seals to activate, making it faster in comparison as a result. This also means that even if one is incapacitated, the other remains completely unaffected and as neither can truly be regarded as the original, the duplicates cannot be dispelled like normal clones.
At first, I thought only Muu could use this technique. But I was wrong.
Zetsu could do it (basically the same).
Juubi did it (though in a different manner)
So, a powerful shinobi like Madara without a doubt could do it too.
Madara, after observing Muu's Fission jutsu, comes up with a grand plan.
Accordingly, Madara acquired the Fission technique, either by observing Muu during their fight or by gaining info on the jutsu through Muu. (My bet is on the latter).
Then, Madara set out for one final clash against his rival/foe/friend Hashirama Senju in order to obtain his DNA. When Madara realized that Shinsuusenju was too powerful for him. He could have switched to his Plan B i.e., The Fission Technique.
(Remember the sole purpose was to obtain Hashi's DNA and get out of it alive
. Whether He won or lost didnt matter). I also think that Madara actually wanted Hashirama to believe that he's dead.
If Madara could take down Hashirama then it was all good. But incase, Hashi was too strong, the Fission Jutsu would be the back-up plan.