What free tools can I use to plot complex functions on the complex plane? I would like to plot $y = e^{ix}$ to view the imaginary unit circle and then change different parameters to see what happens. I am having trouble getting wolfram alpha to do this though. Is there a good way?
First you must define your complex function  as a curve in $\Bbb R^3$ using a parameter, by example $t$, and separating each coordinate.

In our case we have that

$$f(x):=e^{ix}=\cos (x)+i\sin (x)$$

then we can transform the above function in a parametric curve in $\Bbb R^3$ writing

$$f(t)=t\cdot{\bf i}+\cos(t)\cdot{\bf j}+\sin(t)\cdot{\bf k}$$

Using Geogebra we can write in the algebra view

Curve[t,cos(t),sin(t),t,-5,5]


and the graph can be viewed in the 3DView tab. Indeed Geogebra is an extraordinary tool by it simplicity and portability, we can define applets easily as this (with little work we can add text boxes or buttons for any kind of interactivity!)

Using the Wolfram language (in wolfram alpha or the Wolfram programming lab) we can write

ParametricPlot3D[{t,Cos[t],Sin[t]},{t,-5,5}]


And with a very similar code we can write in SageMathCell

t = var('t'); parametric_plot3d([t,cos(t),sin(t)],(t,-5,5), aspect_ratio=[1,1,1], zoom=1.5)


(the result can be seen here).

There are a lot of different tools to graph online, by example I discovered today the library plotly that can be used in many programming languages and online too! The homepage is full of tutorials for any kind of plot, in our case we have a tutorial for a curve in $\Bbb R^3$ here.
Try https://treena.org/lesson/complex-grapher
It should do most of what you want

disclaimer: I helped build it.