I've been given the above circuit and asked to find the equivalent impedance, however after drawing it half a dozen different ways I cannot find a way to simplify it. I can simulate it, and I get (what I think is the right answer) of -0.934-j1.467 ohms, but I'd like to know if there's a way to do it manually, and if there's some useful trick that I'm missing out on here. Or maybe its something simple!
As John D points out: you can do a Delta to Y conversion. Or Y to Delta. That is where you take a set of three resistors (or impedances) and convert them to three other values.
His link points to the wikipedia where they use all resistors. But instead of R you can use a Z and it will still work.
In the schematic above you e.g. take Z2, Z3 and Z5 (That is a Y-formation or T-formation) and make them into a Delta Za, Zb Zc.
I assume you can take it from there.
Looks like you have a three branch parallel circuit with a bridge.
Google how to solve the resistance of a bridge circuit and apply it to your circuit. Chances are it won't be using caps and inductors but you seem to already know about how to calculate the impedance of a basic circuit so its just more of the same thing.
I found this link that walks you thru a simple one but I'm sure you can find others that might be better.