It depends on what kinds of things you can store on the stack.
If each item on the stack is of unbounded size, you can simulate a two-counter machine, which is Turing-complete. Therefore, in this case, you can achieve Turing-completeness with just the ability to access the top two items on the stack.
If items are bounded-size, restricting to access a bounded number of top-most items of the stack gives something equivalent in power to a PDA. (For instance, suppose all you can do is access the top 5 items of the stack, and use that together with a finite control to transition to a new state of the finite control and either push/pop. Then grouping the items into 5-tuples, we can build an equivalent machine where you only need to access the topmost item. That machine is a PDA.) Therefore, in this case, it's not Turing complete with access to any finite number of items at the top of the stack.