Octave (212 143 bytes)

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;end;imagesc(m);colormap([hsv(128)(1+mod(0:79:7900,128),:);0,0,0])

With whitespace:

[x,y] = meshgrid(-2:.01:2);
z = c = x + i*y;
m = 0*e(401);
for n=0:99
    m += abs(z)<2;
    z = z.^2 + c;
end
imagesc(m)
colormap([hsv(128)(1+mod(0:79:7900,128),:);
          0,0,0])

Output:

This would likely work on Matlab too. (Last time I used Matlab, it did not support +=, so fixing that might add one byte. But that was a long time ago.)

My first answer (212 bytes):

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;endfor;t=[0*e(1,7);2.^[6:-1:0]];[s{1:7}]=ndgrid(num2cell(t,1){:});t=1+sum(cat(8,s{:}),8);imagesc(m);colormap([hsv(128)(t(:),:);0,0,0])

Octave (212 136 bytes)

(Now including some ideas due to @ChrisTaylor.)

[y,x]=ndgrid(-2:.01:2);z=c=x+i*y;m=c-c;for n=0:99;m+=abs(z)<2;z=z.^2+c;end;imagesc(m);colormap([hsv(128)(1+mod(0:79:7890,128),:);0,0,0])

With whitespace:

[y,x] = ndgrid(-2:.01:2);
z = c = x + i*y;
m = c-c;
for n=0:99
    m += abs(z)<2;
    z = z.^2 + c;
end
imagesc(m)
colormap([hsv(128)(1+mod(0:79:7900,128),:);
          0,0,0])

Output:

To convert to Matlab, change "m+=abs(z)<2" to "m=m+(abs(z)<2)". [+3 bytes]

To make the aspect ratio 1:1, add ";axis image". [+11 bytes]

My first answer (212 bytes):

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;endfor;t=[0*e(1,7);2.^[6:-1:0]];[s{1:7}]=ndgrid(num2cell(t,1){:});t=1+sum(cat(8,s{:}),8);imagesc(m);colormap([hsv(128)(t(:),:);0,0,0])

Octave - 212

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;endfor;t=[0*e(1,7);2.^[6:-1:0]];[s{1:7}]=ndgrid(num2cell(t,1){:});t=1+sum(cat(8,s{:}),8);imagesc(m);colormap([hsv(128)(t(:),:);0,0,0])

With whitespace:

[x,y] = meshgrid(-2:.01:2);
z = c = x + i*y;
m = 0*e(401);
for n=0:99
    m += abs(z)<2;
    z = z.^2 + c;
endfor
t = [ 0*e(1,7); 2.^[6:-1:0] ];
[s{1:7}] = ndgrid(num2cell(t,1){:});
t = 1 + sum(cat(8,s{:}), 8);
imagesc(m)
colormap([hsv(128)(t(:),:); 0,0,0])

Output:

This would likely work on Matlab too. (Last time I used Matlab, it did not support +=, so fixing that might add one byte. But that was a long time ago.)

Octave (212 143 bytes)

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;end;imagesc(m);colormap([hsv(128)(1+mod(0:79:7900,128),:);0,0,0])

With whitespace:

[x,y] = meshgrid(-2:.01:2);
z = c = x + i*y;
m = 0*e(401);
for n=0:99
    m += abs(z)<2;
    z = z.^2 + c;
end
imagesc(m)
colormap([hsv(128)(1+mod(0:79:7900,128),:);
          0,0,0])

Output:

This would likely work on Matlab too. (Last time I used Matlab, it did not support +=, so fixing that might add one byte. But that was a long time ago.)

My first answer (212 bytes):

[x,y]=meshgrid(-2:.01:2);z=c=x+i*y;m=0*e(401);for n=0:99;m+=abs(z)<2;z=z.^2+c;endfor;t=[0*e(1,7);2.^[6:-1:0]];[s{1:7}]=ndgrid(num2cell(t,1){:});t=1+sum(cat(8,s{:}),8);imagesc(m);colormap([hsv(128)(t(:),:);0,0,0])