Does a filter add oscillations to a signal? this is a more general version of a question which i asked previously.

From what I understand, the purpose of a filter is to change the amplitude of a specific frequency band.

As I went through the analytic calculation of a filtered (2nd order) signal I observed that the output signal contains an oscillating component (a sinusoid, with the frequency being the imaginary part of the complex conjugate pole pair).

A complex pole (in the left half plane) does correspond to a damped oscillation. I really cannot understand how the addition of a specific frequency to the signal conforms with the job of a filter. Thanks for your insights!

EDIT: 

The filter I am considering is a second order low pass Bessel Filter. The input signal is an exponential decay. I tried to calculate the output signal and observed there was a exponentially decaying cosine involved with its frequency being the imaginary part of the two poles. It really surprised me since I did not expect an oscillation in a filtered signal, which it did not express before.

I do understand, that this sinusoid was present in the signal already before filtering, it just struck me that it appears in the filtered signal as a single term.
A linear filter can increase the magnitude of frequency spectra already present in a signal.  Note that the Fourier Transform of any (reasonable) waveform or signal (including ones that do not look at all like they are oscillating) will decompose it into sinusoids (which do oscillate).  Amplify any one frequency, and an oscillation may start to stand out and make it look like it just appeared; but a lower amplitude form of that spectra was already hidden in the input signal to start with.  

It wasn't added.  It was amplified.  Which is possible with an LTI filter.