. . .
But if you're singing pop or jazz, make the accompaniment as simple as possible (like slow chords - listen to Lady Gaga's Born This Way, for instance) so you can concentrate on your vocal production - but remember that some of the most famous pop and jazz singers in history have really atrocious voices, and I'm not talking about their requirement for autotune ......(never mind the voice, it's the thought - or message - that counts whistle).
I sing in a classical choir, and do percussion (and occasionally synth) and sing in a chant group.
For a long time, I could lead a chant, or play, but not both at the same time. I've gotten better at playing and leading singing, after absorbing bennevis's lesson:
. . . Your attention should be focused on your voice, not your fingers.
You must learn the piano part until it's automatic. If that requires simplifying it down to two-handed arpeggios, or a bass line and block chords, do that. Complexities in the piano part should be restricted to when the singer (you) is quiet -- between phrases, or between verses.
When I started singing, I was hoping to play and sing Schubert leider. Maybe in the next lifetime . . .
My teacher could play piano while singing jazz standards, but she had 40 years of experience, starting young.