Usefulness was never seriously questioned. Complex numbers are useful, for example, if you want to make Cardano's formula work. And this is how they were invented.
They are useful for other mathematical problems, and since the beginning of 19th century, in physics too.
What was questioned sometimes is their justification, existence. These questions gradually disappear when their geometric interpretation became a common knowledge,
which happened in the beginning of 19th century. But still some mathematicians of
19th century (Chebyshev, for example) distrusted them and tried to avoid them.
I read somewhere (don't remember where) that students of Ecole Polytechnique once in the beginning of 19th century made a riot protesting teaching complex numbers (on the grounds that they are useless in practical applications),
if this story is true, these were just poorly educated students. Serious mathematicians never doubted that they are useful.