  We perform a bifurcation analysis of a two-dimensional magnetic
Rayleigh--B\'enard problem using a numerical technique called deflated
continuation. Our aim is to study the influence of the magnetic field on the
bifurcation diagram as the Chandrasekhar number $Q$ increases and compare it to
the standard (non-magnetic) Rayleigh--B\'enard problem. We compute steady
states at a high Chandrasekhar number of $Q=10^3$ over a range of the Rayleigh
number $0\leq \Ra\leq 10^5$. These solutions are obtained by combining
deflation with a continuation of steady states at low Chandrasekhar number,
which allows us to explore the influence of the strength of the magnetic field
as $Q$ increases from low coupling, where the magnetic effect is almost
negligible, to strong coupling at $Q=10^3$. We discover a large profusion of
states with rich dynamics and observe a complex bifurcation structure with
several pitchfork, Hopf, and saddle-node bifurcations. Our numerical
simulations show that the onset of bifurcations in the problem is delayed when
$Q$ increases, while solutions with fluid velocity patterns aligning with the
background vertical magnetic field are privileged. Additionally, we report a
branch of states that stabilizes at high magnetic coupling, suggesting that one
may take advantage of the magnetic field to discriminate solutions.
