What I'm wondering about, cannot we by density of $\Bbb C$-valued points reduce the statements about connectedness & irreducibility of the fibres reduce to analogous statements about fibres wrt $p(\Bbb C)$? If yes, cannot we reasoning that way also deduce not only that all fibres are connected, but even irreducible?
I know that the latter cannot be the case, as there are concrete counterexamples with families where general irreducible fibers degenerate to reducible fibres. But this raises then the question where the in following presented "argumentation" would break down?

My point was why cannot we also argue via Ehresmann Lemma? For proof see, eg Prop 6.2.2, Complex Geometry, Huybrechts.
If not, why? if yes, wouldn't this imply even stronger statement that all fibres are irreducible assuming fibres over $U$ are? (..see below why I think so)

What I'm wondering about, cannot we by density of $\Bbb C$-valued points reduce the statements about connectedness & irreducibility of the fibres reduce to analogous statements about fibres wrt $p(\Bbb C)$? If yes, cannot we reasoning that way also deduce not only that all fibres are connected, but even irreducible?

I know that the latter cannot be the case, as there are concrete counterexamples with families where general irreducible fibers degenerate to reducible fibres. But this raises then the question where the in following presented "argumentation" - which if I'm not overlooking something implies even that the fibres irreducible - would concretely break down?

My point was why cannot we also argue here via Ehresmann Lemma? For proof see, eg Prop 6.2.2, Complex Geometry, Huybrechts.
If not, why? if yes, wouldn't this imply even stronger statement that all fibres are irreducible assuming fibres over $U$ are? (..see below why I think so)

This raises Two Questions:

This raises two questions:

This requires some clarification as Ehresmann works in differential geometric setting, and says that such map $f(\Bbb C)$ as above is locally a trivial fibration in differential geometric setting.
Attention: As these local trivializations are given not complex analytically, but differential geometrically, ie there exist open $V \subset X(\Bbb C)$ (wrt smooth topol) such that $(p(\Bbb C))^{-1}(V)$ is diffeomorphic to $(p(\Bbb C))^{-1}(x)) \times V$. Note, that this completely forgets about complex structure of the maps.

Clearly, this requires some clarification as Ehresmann works in differential geometric setting, and says that such map $f(\Bbb C)$ as above is locally a trivial fibration in differential geometric setting.
Attention: As these local trivializations are given not complex analytically, but differential geometrically, ie there exist open $V \subset X(\Bbb C)$ (wrt smooth topol) such that $(p(\Bbb C))^{-1}(V)$ is diffeomorphic to $(p(\Bbb C))^{-1}(x)) \times V$. Note, that this completely forgets about complex structure of the maps.