Discrete Wavelet Transform (Daubechies wavelet) of an array complex numbers

Say, I have a signal represented as an array of real numbers y = [1,2,0,4,5,6,7,90,5,6]. I can use Daubechies-4 coefficients D4 = [0.482962, 0.836516, 0.224143, -0.129409], and apply a wavelet transform to receive high- and low-frequencies of the signal. So, the high frequency component will be calculated like this:

high[v] = y[2*v]*D4[0] + y[2*v+1]*D4[1] + y[2*v+2]*D4[2] + y[2*v+3]*D4[3], 

and the low frequency component can be calculated using other D4 coefs permutation.

The question is: what if y is complex array? Do I just multiply and add complex numbers to receive subbands, or is it correct to get amplitude and phase, treat each of them like a real number, do the wavelet transform for them, and then restore complex number array for each subband using formulas real_part = abs * cos(phase) and imaginary_part = abs * sin(phase)?

To handle the case of complex data, you're looking at the Complex Wavelet Transform. It's actually a simple extension to the DWT. The most common way to handle complex data is to treat the real and imaginary components as two separate signals and perform a DWT on each component separately. You will then receive the decomposition of the real and imaginary components.

This is commonly known as the Dual-Tree Complex Wavelet Transform. This can best be described by the figure below that I pulled from Wikipedia:

^{Source: Wikipedia}

It's called "dual-tree" because you have two DWT decompositions happening in parallel - one for the real component and one for the imaginary. In the above diagram, g0/h0 represent the low-pass and high-pass components of the real part of the signal x and g1/h1 represent the low-pass and high-pass components of the imaginary part of the signal x.

Once you decompose the real and imaginary parts into their respective DWT decompositions, you can combine them to get the magnitude and/or phase and proceed to the next step or whatever you desire to do with them.

The mathematical proof regarding the correctness of this is outside the scope of what we're talking about, but if you would like to see how this got derived, I refer you to the canonical paper by Kingsbury in 1999 in the work Image Processing with Complex Wavelets. Pay close attention to the noise filtering of images using the CWT - this is probably what you're looking for.