  We propose a new multi-dimensional discrete-time quantum walk (DTQW), whose
continuum limit is an extended multi-dimensional Dirac equation, which can be
further mapped to the Schr\"{o}dinger equation. We show in two ways that our
DTQW is an excellent measure to investigate the two-dimensional (2D) extended
Dirac Hamiltonian and higher-order topological materials. First, we show that
the dynamics of our DTQW resembles that of a 2D Schr\"{o}dinger harmonic
oscillator. Second, we find in our DTQW topological features of the extended
Dirac system. By manipulating the coin operators, we can generate not only
standard edge states but also corner states.
