Scales to Play Over Amaj7#11
Every chord is a subset of several scales: any scale that contains all of a chord's tones fits over it. Pick a chord to see the matching scales from its own root, the major keys the chord belongs to, and full fretboard diagrams for every scale on guitar.
An Amaj7#11 chord contains the notes A – C# – D# – E – G#. Any scale that includes every one of those 5 tones fits over it — from the root A, that means A Lydian. It is the only scale in the site's catalog that holds every one of those tones.
Amaj7#11 is also diatonic to one major key: E major (relative minor C#m). In a song from one of those keys you can also solo with the key's own major scale or its relative minor pentatonic across the whole progression, not just this chord.