It seems very probable to me that the method is relatively complete, even if reality is sufficiently complex that people will be able to keep getting interesting results by using the scientific method even should we manage to live for the rest of eternity.
The scientific method is simple really – it is to question everything, then design experiments to test which of the available explanations best fit the data given by the experiments, then repeat.
And the world seems to be very complex.
Our best explanatory framework for physical reality (Quantum mechanics) deals fundamentally with probabilities. So one has to give up the classical notion of Truth that most of us start with, and end up with sets of explanations that deliver contextually useful probabilities (even if we are never entirely certain we are in an appropriate context 😉 ).
Exactly how we generate variants on ideas that seem to us to be likely enough to be useful enough to be worth the effort of testing is itself one of the more fascinating areas of inquiry – and the AI community is leading the way in many aspects at present. And it seems likely to remain as much art as science.
So I suspect that the scientific method will be with us for a very long time (as long as we manage to survive); even if the idea of knowledge that we gain from it is a very different sort of thing from the one we are taught in high school.