I have a complex signal generated by an impedance analyzer.
In fact you have two signals, and it depends on what you want to achieve, but usually you would just filter both signals (the real and the imaginary part) with the same (real-valued) low pass filter. So you either need two (identical) low pass filters, or you filter both signals sequentially with the same filter.
In the general case, filtering a complex signal $x(t)=x_R(t)+jx_I(t)$ with a complex impulse response $h(t)=h_R(t)+jh_I(t)$ requires four real-valued filtering operations (just like complex multiplication requires four real-valued multiplications):
$$\begin{align}y(t)&=(x\star h)(t)\\&=(x_R\star h_R)(t)-(x_I\star h_I)(t)+j\left\{(x_R\star h_I)(t)+(x_I\star h_R)(t)\right\}\end{align}\tag{1}\\$$
In your case $h_I(t)$ is zero, so you're left with just two filtering operations.
$$\begin{align}y_1(t)&=(x_R\star h_R)(t)\\ y_2(t)&=(x_I\star h_I)(t)\\ y_3(t)&=((x_R+x_I)\star (h_R+h_I))(t)\\ y(t)&=y_1(t)-y_2(t)+j\left\{y_3(t)-y_1(t)-y_2(t)\right\}\end{align}\tag{2}\\$$
I've added some graphics that help illustrate the other good answers provided here. Not shown are all permutations such as the cases with imaginary only outputs but this should be sufficient for providing an intuitive sense of the various options and there implications.
As Matt details in his answer, these variants are all clear by starting with understanding the implementation of a full complex multiplier (thus each of the coefficients will follow this pattern with the datapath as detailed in the diagrams below):
$$(x_I+jx_Q)(h_I+jh_Q) = (x_I h_I - x_Qh_Q) + j (x_Ih_Q + x_Qh_I)$$
This results in case 1 below for complex inputs, complex filter and complex outputs. Convolution in the filter would follow the similar computation with dot and cross products of the real and imaginary components of the input with the real and imaginary components of the filter coefficients. By restricting the input, the filter or the output to real only, we get the other combinations I show below, with the resulting spectrums depicting the behavior for each case:
Paraphrasing Matt L.'s answers in the comments:
Matt: "[Design is] different if you want to design a complex filter, i.e. a filter with a complex-valued impulse response. But this is only necessary if for some reason you want a non-symmetrical frequency response."
Matt: "[Otherwise] You design the (real-valued) filter as you would normally, and then you apply it to the real part as well as to the imaginary part of the input signal."
Me: Since I don't want to filter the negative frequencies differently than the positive ones, I'll just create the appropriate real filter and then apply it to the separate real and imaginary data streams. I don't need to worry about the cross terms in eqn. 1 because Hi = 0 by definition.
The term 'complex' is in the realm of mathematics. It's hypothetical. Complex 'data' is treated as information. So when we transmit this mathematical (hypothetical) data in the real world, we need a system that gets that math data across - in some way. So, if the impedance analyser is generating a 'complex' signal, then this signal could be a real-valued signal, except it might involve the sum of TWO sinusoidal signals, each sinusoid having identical frequency and phase difference between these sinusoids being 90 degrees. In this way, the actual signal is real-valued.
If the appropriate mathematical and hardware processing trickery is applied at the receiving side, then it becomes possible to grab (recover) the 'real' and 'imaginary' components from the summed-sinusoid (composite) waveform.
Depending on what sort of processing you want to do (ie. just process samples of the composite waveform only and have no further need for real/imaginary components, or process individually real and imaginary components) will determine whether 1 filter is enough, or whether 2 filters are needed.
