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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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Modal logics which have an algebraic semantics but not a Kripke semantics
I guess the logic your colleague mentioned is quasi-normal modal logic. The book Modal Logic by Alexander Chagrov and Michael Zakharyaschev contains this. But it indeed can have a non-standard Krikpe …