Concept:
Slow sand filters remove suspended impurities as well as pathogenic microorganisms through mechanical straining, adsorption, and biological action.
A biological layer called the Schmutzdecke develops on the top of the filter bed, which is mainly responsible for the excellent bacteriological purification.
Step 1: Understand the filtration mechanism.
The Schmutzdecke layer removes bacteria by biological action and adsorption.
Hence, slow sand filters produce water of very high bacteriological quality.
Step 2: Compare with rapid sand filters.
Rapid sand filters mainly remove suspended solids and require chlorination for effective disinfection.
Therefore,
\[
\boxed{\text{Slow sand filters produce water of the highest bacteriological quality}.}
\]
Hence, the correct option is
\[
\boxed{(A)\;\text{Slow sand filter}.}
\]