Hi thomas,
From the theory, zero makes the transfer function to zero value, and zero produces +20db gain and positive or negative 90 degree phase shift.
I think, your understanding of the meaning of the complex frequency variable s is not yet good enough - and the difference to the real-world frequency w (omega). Remember: The transfer function H(s) turns into the so called frequency response H(jw) if you replace s by jw.
You are right the transfer function H(s) can go to zero for a certain value of s. For example, think of a simple highpass of 2nd order. Buth this does not mean that you can measure zero volts with a function generator. The zero is located in the left hand part of the complex plane. Only if the zero is placed directly on the jw axis it is identical to a "real" zero because H(jw)=0.
Secondly, a complex zero does not "produce" a gain of +20 dB.( It produces a slope increase in the BODE plot that has an asymptote of +20 dB/decade.) Correction: It produces a slope change in the bode plot by an asymptotic value of +20 dB/decade.