Is there any application for $\Re(f(z)) + \Im(f(z))$?

Question

I was messing around with inverse Laplace transformations of functions with single complex poles and was looking at step responses for various systems by adding the real and imaginary parts as $\Re(f(z)) + \Im(f(z))$ and was wondering if this has any real applications in complex analysis. Results for $f(z) = \mathscr{L}^{-1}\big(\frac{1}{s^n(s+i)}\big)$ with positive integers $n$ always gave nicely behaved sums of exponentials and sines/cosines as expected, but does this provide any meaningful insight into the behavior of complex functions above what could be gained from more common methods of analysis?