#Excel VBA, 251 bytes

<!-- language-all: lang-vb -->

    Sub m()
    D=99
    For x=1To 4*D
    For y=1To 4*D
    p=0
    q=0
    For j=1To 98
    c=2*p*q
    p=p^2-q^2-2+(x-1)/D
    q=c+2+(1-y)/D
    If Sqr(p^2+q^2)>=2Then Exit For
    Next
    j=-j*(j<99)
    Cells(y,x).Interior.Color=Rnd(-j)*2^20*j/D
    Next
    Next
    Cells.RowHeight=9
    Cells.ColumnWidth=1
    End Sub

Output:

[![Output with D = 99][1]][1]

I made a version that did this a long time ago but it had a lot of extras like letting the user pick the basic color and easy-to-follow math. Golfing it way down was an interesting challenge. The `Color` method uses `2^20` as a means to get a wide range of colors since the valid colors are `0` to `2^24`. Setting the exponent to `20` gave higher contrast areas.

Explanation / Auto-Formatting:

    Sub m()
        
        'D determines the number of pixels and is factored in a few times throughout
        D = 99
        For x = 1 To 4 * D
            For y = 1 To 4 * D
                'Test to see if it escapes
                'Use p for the real part and q for the imaginary
                p = 0
                q = 0
                For j = 1 To 98
                    'This is a golfed down version of complex number math that started as separate generic functions for add, multiple, and modulus
                    c = 2 * p * q
                    p = p ^ 2 - q ^ 2 - 2 + (x - 1) / D
                    q = c + 2 + (1 - y) / D
                    If Sqr(p ^ 2 + q ^ 2) >= 2 Then Exit For
                Next
                
                'Correct for no escape
                j = -j * (j < 99)
                
                'Store the results
                'Rnd() with a negative input is deterministic
                'This is what gives us the distinct color bands
                Cells(y, x).Interior.Color = Rnd(-j) * 2 ^ 24 * j / D
                
            Next
        Next
        
        'Resize for pixel art
        Cells.RowHeight = 9
        Cells.ColumnWidth = 1
        
    End Sub


----------

I also played around with `D=999` and `j=1 to 998` to get a much larger and more precise image. The results are irrelevant to the challenge because they're way too large but they *are* neat.

[![D=999][3]][3]


  [1]: https://i.sstatic.net/jcPqJ.png
  [2]: https://i.sstatic.net/OSPY9.png
  [3]: https://i.sstatic.net/FHIHY.jpg