The maj7(#5) chord can be seen as augmented chord with an additional major 7th. However, it is uncommon to call it aug(maj7).
A Cmaj7(#5) has the notes C (1), E (3), C (#5) and B (maj7).
Since E, G# and B togeher make a major chord, Cmaj7(#5) can also be written as slash chord E/C („E over C”), a E major chord over a C bass note.
In a line cliché usualy just one note is moving chromatically from chord to chord. In the chord progression CCmaj7(#5) C6 it's two notes, but the effect is vitually the same.
The augmented 5th (#5) creates a tension that can be used for a delay of resolution, e.g. in the chord progression Dm7 G7(13) Cmaj7(#5) C6.
Adding a maj7(#5) can give the next chord more of a resolving character, e.g. Cmaj7 Cmaj7(#5) Am7.
In the progression Cmaj7 Cmaj7(#5) Dm7 you may also feel the chromatic transition between the chords.
Thoughts for geeks: The maj7(#5) chords tendency to resolve to a m7 chord a minor 3rd below, like Cmaj7(#5) to Am7, makes sense when writing it as slash chord. Cmaj7(#5) = E/C. Since the bass note C can somehow be seen as b6/b13 of C, you can see E/C as E7(b13) with the 7th not played. E7(b13) would be the minor dominant of Am7 and therefore perfectly resolves to it.
As a diatonic chord (the chords you get by stacking 3rds over each note of a scale) you can NOT find the maj7(#5) in the major or minor scale, but it can be deduced that way from the third degree of the harmonic minor and also melodic minor scale.