I'm curious what you mean by complementary workers.
In the example of immigrants doing hard manual labor, especially agricultural work, who are the complementary workers? I'd infer it's workers like the people who transport crops around or operate the machinery to plant / grow the crops, etc.
Is that close to the meaning you intended?
Immigration indeed lowers wages for competing workers, but it also increases wages for complementary workers and generates surplus to owners of capital – which in the long-term constitute a net gain to all of society, but at a minimum does constitute a net loss to workers at large.
Research suggests negative wage effects are concentrated among low-skilled workers.
Immigration can indeed have disruptive effects, but it can be managed, and its net gains for the whole of society are fairly well established.