We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree delta(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphs H with delta(H) >= n/2 - C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpinski, Rucinski and Szymanska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant epsilon > 0, determines whether or not a 4-uniform hypergraph H on n vertices with delta(H) >= n/2 - epsilon * n contains a Hamilton 2-cycle. This demonstrates that some looser Hamilton cycles exhibit interestingly different behaviour compared to tight Hamilton cycles. A key part of the proof is a precise characterisation of all 4-uniform hypergraphs H on n vertices with delta(H) >= n/2 - epsilon * n which do not contain a Hamilton 2-cycle; this may be of independent interest. As an additional corollary of this characterisation, we obtain an exact Dirac-type bound for the existence of a Hamilton 2-cycle in a large 4-uniform hypergraph.