#Excel VBA, <s>251</s> <s>246</s> <s>224</s> <s>223</s> 221 bytes

<sub><sup>Saved 5 bytes thanks to ceilingcat</sup></sub>
<sub><sup>Saved 23 bytes thanks to Taylor Scott</sup></sub>

<!-- 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 p^2+q^2>=4Then Exit For
    Next
    j=-j*(j<D)
    Cells(y,x).Interior.Color=Rnd(-j)*1E6*j/D
    Next y,x
    Cells.RowHeight=48
    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 `1E6` as a means to get a wide range of colors since the valid colors are `0` to `2^24`. Setting it to `10^6` gave nice 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 p ^ 2 + q ^ 2 >= 4 Then Exit For
            Next
    
            'Correct for no escape
            j = -j * (j < D)
    
            '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) * 1000000# * j / D
    
        Next x, y
    
        'Resize for pixel art
        Cells.RowHeight = 48
    
    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