  For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space
$\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined
as \begin{align*} \Lambda_{\varepsilon}^{S}(T) := \sigma_S (T) \bigcup \left \{
q \in \mathbb{H}\setminus \sigma_S(T):\; \|\Delta_{q}(T)^{-1}\| \geq
\frac{1}{\varepsilon} \right\}, \end{align*} where $\mathbb{H}$ denotes the
division ring of quaternions, $\sigma_S(T)$ is the $S$-spectrum of $T$ and
$\Delta_q(T)= T^2-2 \text{Re}(q)T+|q|^2I$. This is a natural generalization of
pseudospectrum from the theory of complex Hilbert spaces. In this article, we
investigate several properties of the pseudo $S$-spectrum and explicitly
compute the pseudo $S$-spectra for some special classes of operators such as
upper triangular matrices, self adjoint-operators, normal operators and
orthogonal projections. In particular, by an application of $S$-functional
calculus, we show that a quaternionic operator is a left multiplication
operator induced by a real number $r$ if and only if for every $\varepsilon>0$
the pseudo $S$-spectrum of the operator is the circularization of a closed disc
in the complex plane centered at $r$ with the radius $\sqrt{\varepsilon}$.
Further, we propose a $G_1$-condition for quaternionic operators and prove some
results in this setting.
