What type of cipher or encryption method is this called?
Adding the previous block to the plaintext before encryption is what the CBC mode of operation does. So you're essentially treating the Caesar cipher as a block-cipher and using it in CBC mode.

This adds no security relative to a plain Caesar cipher, since the attacker can simply undo the addition before proceeding to break Caesar as usual.


  the majority of the well known ciphers


That property is shared by substitution ciphers, but not by Vigenere. Modern ciphers don't have any such obvious weaknesses either if you use them correctly.