The Intel architecture is already Turing complete when you just use MOV instructions: https://github.com/xoreaxeaxeax/movfuscator. Of course, you don't even need instructions at all: https://news.ycombinator.com/item?id=5261598
No physically existing architecture is Turing-complete, since every CPU can (by physics) only access a finite amount of memory, which means that its state space is finite, in opposite to the infinite state space of a Turing machine.