$S$ cannot be a 3-dimensional real hyperboloid unless $\psi$ is of the same form as $\phi$, though it can be a 2-dimensional real hyperboloid*.
Excluding the possibilities achievable when $\psi$
is a real quadratic form, and working up to similarity of $\psi$ over $\mathcal{R}$,
$S$ is one of the following:

$\quad\bullet\quad\emptyset$

$\quad\bullet\quad$ a point

$\quad\bullet\quad$ a 2-dimensional quadric surface in the form of a cone, an ellipsoid, a hyperboloid of two sheets*, or a hyperboloid of one sheet.

$\quad\bullet\quad$ the Cartesian product of a circle with two intersecting lines

$\quad\bullet\quad$ the Cartesian product of a hyperbola with two intersecting lines

PROOF:

By the inertia theorem, up to similarity over $\mathbb{R}$,
there exist $a,b,c,d\in\{\pm 1,\pm i\}$
such that $\psi(w,x,y,z)=aw^2+bx^2+cy^2+dz^2$.
In the case where $a,b,c,d\in\{\pm 1\}$,
we have the usual quadrics obtainable by real quadratic forms on $\mathbb{R}^4$,
and in this case $S$ is a hyperboloid model if and only if $\psi$
takes the form of $\phi$, contributing nothing new.
So assume that not all of $a,b,c,d$ are $\pm 1$, and we will work up to permutation of coordinates.

Suppose $a,b,c,d\in\{\pm i\}$.
Then $\psi=i\phi'$
where $\phi'$ is a real quadratic form, therefore $i\phi'(w,x,y,z))$
is purely imaginary, giving $\forall w,x,y,z\in\mathbb{R}:\Re(\psi(w,x,y,z))=0\neq-1$.
So $S=\emptyset$.

Suppose $a\in\{\pm1\}$ and $b,c,d\in\{\pm i\}$.
Then $\Re(\psi(w,x,y,z))=\pm w$ and $\Im(\psi)=i\phi'$
where $\phi'$ now is a real quadratic form on $\mathbb{R}^3$.
So we require $w=\pm 1$, and $\phi'(x,y,z)=b'x^2+c'y^2+d'z^2=0$
(where $b'$ is the real number $\Im(b)$, etc.) which gives either $x=y=z=0$ (when the signs agree)
or $(x,y,z)$ lie on a two-dimensional cone (when the signs do not agree).
So $S$ is either the point $(\pm1,0,0,0)$, or a cone lying in the copy of $\mathbb{R}^3$ at $w=1$ or $w=-1$.

Suppose $a,b\in\{\pm1\}$ and $c,d\in\{\pm i\}$.
If $a,b=1$ then $\Re(\psi(w,x,y,z))=w^2+x^2\neq-1$, making $S=\emptyset$.
If $a,b=-1$ then $\Re(\psi(w,x,y,z))=-w^2-x^2=-1$ $\Longrightarrow(w,x)$ lie on a unit circle.
If $a=1, b=-1$ then $\Re(\psi(w,x,y,z))=w^2-x^2=-1\Longrightarrow(w,x)$ lie on a regular hyperbola.
But for $\phi(w,x,y,z)$ to be $-1$ we need $\Im(\phi(w,x,y,z))=0$.
If $c=d=\pm i$ then this forces $y=z=0$.
If $c=i$ and $d=-i$ then this forces $y^2=z^2$,
i.e. $(y,z)$ lie on two perpendicular lines.
Combining this with the observations about the real part, we have that $S$ is either $\emptyset$, or a cartesian product of a curve with a pair of perpendicular lines, where the curve is either a circle or a hyperbola.

Lastly, suppose $a,b,c\in\{\pm1\}$ and $d\in\{\pm i\}$. Then to get $\phi(w,x,y,z))=-1$ we require that $z=0$, and that the solution for $(w,x,y)$
to $aw^2+bx^2+cy^2=-1$ is either empty, an ellipsoid, a hyperboloid of one sheet, or a hyperboloid of two sheets, as determined by the signs.