I have a function $R to C$, I want to plot it as the way we can plot parametric equations in the $R^2$. How to do it?

ReImPlot[2 t + I t^2, {t, -[Pi], [Pi]}]

That is a new built-in in V12 and newer: ReImPlot

ReImPlot[{ArcSin[x], ArcCos[x]}, {x, -4, 4}, 
 PlotLabels -> "Expressions"]

The input that suffices for WolframAlpha is simpler

WolframAlpha["2 t+I t^2 plot -pi to pi"]

WolframAlpha["{ArcSin[x],ArcCos[x]} plot -4 to 4"]

has some unexpected behavior with the desired interval. It does not show the continuation.

An introductory example is

ReImPlot[{Sqrt[1 - x^2], -Sqrt[x^2 - 1]}, {x, -3, 3}]

This separates the branches of roots very neat.

AbsArgPlot[1 + Exp[-Abs[x]] Sin[I Sin[5 x]], {x, -Pi, Pi}, 
 PlotRange -> Full]

AbsArgPlot is an acompanying function suiting the given criteria too.

A Mathematica example is the Nyquist plot:

h = 1/(s - 1/2)^2 /. s -> Exp[I [Omega]]
ParametricPlot[{Re[h], Im[h]}, {[Omega], 0, 2 Pi}]