The most succinct answer to the question why complex numbers are so useful in the analysis of physical and engineering problems was given by Paul Painlevé in 1900: "between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain" (usually this quote is known in Jacques Hadamard's later formulation).

But are complex numbers really essential in physics? The usual argument is based on the Heisenberg's uncertainty principle in the form $[\hat x,\hat p]=i\hbar$. In the words of Paul Dirac (The principles of quantum mechanics, §10, p.35. Stückelberg in his 1960 paper "Quantum Theory in Real Hilbert Space" also provided the similar argument):

One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary parts. But this would involve two measurements or two observations, which would be alright in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another - it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the system and introduce an indeterminacy that will affect the second.

This point of view is further extended by Chen Ning Yang in https://www.worldscientific.com/doi/abs/10.1142/9789814449021_0014 (Square root of minus one, complex phases and Erwin Schrödinger). Yang traces the entry of complex numbers into fundamental physics to Schrödinger's 1922 paper in which he had mentioned the possibility of introducing an imaginary factor into Weyl's 1918 gauge theory. The development of this idea by London, Fock and Weyl lead to the gauge theory of electromagnetism. Yang writes:

The importance of the introduction of complex amplitudes with phases into
physicists' description of nature was not fully appreciated until the 1970s
when two developments took place: (1) all interactions were found to be
some form of gauge field; and (2) gauge fields were found to be related to the mathematical concept of fibre bundles (Wu and Yang, 1975), each fibre being a complex phase or a more general phase. With these developments there arose a basic tenet of today's physics: all fundamental forces are phase fields (Yang, 1983). Thus the almost casual introduction in 1922 by
Schrödinger of the imaginary unit i has flowered into deep concepts that lie at the very foundation of our understanding of the physical world.

Although the quantum mechanics can be formulated in real Hilbert space, such a formulation is redundant and can be always reformulated in the complex Hilbert space, see https://arxiv.org/abs/1611.09029 (Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry, by Valter Moretti and Marco Oppio).

A fascinating history of $\sqrt{-1}$ can be found in the book of Paul Nahin
"An Imaginary Tale: the story of $\sqrt{-1}$ (here is a review of this book by C O'Sullivan: http://iopscience.iop.org/article/10.1088/0143-0807/20/2/013 )