The normal view taken by Haskellers is that the denotations of
Haskell types are CPPOs.
CPPO?
So:
(1) Must be monotone (2) Must be continuous
Could you please define what you mean by those terms in this context?
(Needn't be strict, even though that messes up the resulting category substantially).
I'm not convinced that the category is all that "messed up".
The extra P would stand for "pointed" (has a least element, bottom), this is common in some communities. To me though, a cpo (complete partial order) is closed under directed suprema and the empty set is directed so bottom is already required. The category of cpos in not cartesian closed. For denotational semantics I believe the subcategory of Scott domains are what is usually considered.
Continuous functions on cpos are by definition monotone and they respect directed suprema. Jens