Given the versatility of timed automata a huge body of work has evolved that considers extensions of timed automata. One extension that has received a lot of interest is timed automata with a, possibly unbounded, stack, also called pushdown timed automata (PDTA). While different algorithms have been given for reachability in different variants of this model, most of these results are purely theoretical and do not give rise to efficient implementations. One main reason for this is that none of these algorithms (and the implementations that exist) use the so-called zone-based abstraction, but rely either on the region-abstraction or other approaches, which are significantly harder to implement. In this paper, we show that a naive extension, using simulations, of the zone based reachability algorithm for the control state reachability problem of timed automata is not sound in the presence of a stack. To understand this better we give an inductive rule based view of the zone reachability algorithm for timed automata. This alternate view allows us to analyze and adapt the rules to also work for pushdown timed automata. We obtain the first zone-based algorithm for PDTA which is terminating, sound and complete. We implement our algorithm in the tool TChecker and perform experiments to show its efficacy, thus leading the way for more practical approaches to the verification of timed pushdown systems.