Only works in range -Infinity < x < 1
FullSimplify[ArcCos[x] + I ArcCosh[x], Assumptions -> -Infinity < x < 1]
(* 0 *)
Plot[ArcCos[x] + I ArcCosh[x], {x, -2, 2}, PlotRange -> {{-2, 2}, {-1, 1}},
PlotStyle -> {Black, Thickness[0.02]}]
NumberLinePlot[{-Infinity < x < 1}, {x, -2, 2}, PlotTheme -> "Detailed"]