Orbit diagram

Fixed points: $p_+$ (repelling), $p_-$ (attracting if $c>-\tfrac34$).
2-cycle $q_\pm$ exists when $c<-\tfrac34$ and is attracting if $c>-\tfrac54$.
When $c<-2$: $A_1$ and $A_2$ are the first two steps of the Cantor set construction. Orbit eventually escapes, unless the seed comes from the Cantor set.