I cannot think of any such model, maybe some form of typed lambda calculus? some elementary cellular automaton?This would almost disprove Wolfram's Principle of Computational Equivalence:Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication
Is there a non Turing-complete model of computation whose halting problem is undecidable?
automata theory;computability;turing machines;lambda calculus;universal computation
You can easily build artificial models which are not Turing complete but the halting problem for them is undecidable. E.g. take all TMs that do not halt on anything but $0$.Regarding the statement:You cannot disprove a statement that is not precise enough. Almost none of the words in the statement is well-defined (please provide the definition for them if this is not the case).