I have a a-plane nested in another a-plane.
<a-plane width="10" height="5" color="#FFFFFF">
<a-plane width="10" color="#000080"></a-plane>
</a-plane>
But the result looks like this I also tried it with a nested a-image and had the same result. I also tried to remove the inner plane and add it outer and to the same position. But I've got always the same result.
A-Frame Version: 0.9.2
You've placed two planes in an identical position. The renderer does not know which one should be in front of another (since they are planes and their transforms are identical) - and hence you experience z-fighting.
You need to position one of the planes so that they won't overlap:
<a-plane>
<a-plane position='0 0 -0.001'>
</a-plane>
</a-plane>