Take the complex surface $Z_1 ^2 + Z_2 ^2 + Z_3^2 =1$ in complex 3-space, $\mathbb{C}^3$ and intersect it with the ball $|Z_1|^2 + |Z_2|^2 + |Z_3|^2 \le 1$ to get an explicit 4-manifold with boundary embedded in $\mathbb{C}^3$ whose boundary is $\mathbb{R}P^3$, realized by intersecting the complex surface with the 5-sphere $|Z_1|^2 + |Z_2|^2 + |Z_3|^2 = 1$.

To verify, split $Z_a = x_a + i y_a$ into its real and imaginary parts $x_a, y_a$, do a bit of algebra and see that the hypersurface is diffeomorphic to the tangent bundle of the standard two-sphere $S^2$, that two-sphere being realized within $(x_1, x_2, x_3)$ space by setting the $y_a = 0$. (I've heard this complex surface called the 'hypersphere' or 'complex sphere' or some such.)

Intersecting the hypersurface with the 5-ball $|Z_1|^2 + |Z_2|^2 + |Z_3|^2 \le 1$ is the same as taking the disc bundle of Tim Perutz's construction.

Setting $|Z_1|^2 + |Z_2|^2 + |Z_3|^2 = 1$ within the hypersurface is the $RP^3$ which bounds the 4-manifold.

So, you can take your 4-manifold to be a ``Stein domain'' in standard $\mathbb{C}^3$.

Even more explicitly, with a CR-aside:

After some rescalings, this embedding of $RP^3$ is essentially Rossi's example of an analytic CR structure on the three-sphere which admits no CR embedding into any $C^n$.

Consider the map from $\mathbb{C}^2$ to $\mathbb{C}^3$ given by
$$Z_1 = i[ (z^2 + w^2) + t (\bar z ^2 + \bar w ^2)]$$
$$Z_2 = [ (z^2 - w^2) - t (\bar z ^2 - \bar w ^2)]$$
$$Z_3 = 2 [ zw - t \bar z \bar w]$$
where $t$ is real and $i = \sqrt{-1}$.

A computation shows that
$Z_1 ^2 + Z_2 ^2 + Z_3^2 =-4 t (|z|^2 + |w|^2)^2$
while
$|Z_1|^2 + |Z_2|^2 + |Z_3|^2 = 2(1 + t^2)(|z|^2 + |w|^2)^2.$
It follows that the image of the standard $S^3$ in $C^2$ is the
complex hypersurface
$Z_1 ^2 + Z_2 ^2 + Z_3^2 =-4 t $
intersected with the 5-ball

$|Z_1|^2 + |Z_2|^2 + |Z_3|^2 = 2(1 + t^2)$.

Since the map $(z, w) \to (Z_1, Z_2, Z_3)$
is $2:1$ restricted to $S^3$, its image is $\mathbb{R}P^3$.

Rossi's 'no CR embedding' assertion for $t \ne 0$ relates to the induced CR structure on $S^3$ (via the pull-back of its image's CR structure). When $t \ne 0$
every CR function for this CR structure on $S^3$ is an analytic function of these $Z_i (z,w)$'s, and hence is invariant under the antipodal map $(z,w) \to (-z, -w)$. Thus the CR functions for these `twisted' CR structures on $S^3$ cannot separate (antipodal) points and hence the $S^3$ cannot be CR-embedded.

It is a fun fact that all left-invariant CR structures on $SU(2) = S^3$ arise this way, with the standard CR structure corresponding to $t = 0$.

some references:

H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave
boundary. 1965 Proc. Conf. Complex Analysis (Minneapolis, 1964)
pp. 242–256, Springer, Berlin.

D. Burns, Global behavior of some tangential Cauchy-Riemann equations
in “Partial Differential Equations and Geometry” (Proc. Conf., Park City,
Utah, 1977); Dekker, New York, 1979, p. 51.

E. Falbel, Non-embeddable CR-manifolds and Surface Singularities. Invent.
Math. 108 (1992), No. 1, 49-65.