What are the minimum requirements for a language to be considered Turing Complete?
The machine you describe would be turing complete with the addition of an assignment operator (you need this to take advantage of memory), and at least one comparison operator (which could be folded into the jump, like a jump-if-(not)-zero). (Note that an increment operator is necessary - either as a primitive, or appropriate representations need to be available such that one can roll ones own).

However, there are many turing-complete models of computation with completely different primitive operations, such as lambda calculus, or petri nets.