The term "complex hybrid" applies to all the different orchid genera, not just Paphs., so it is not a synonym of "bulldog".
Primary hybrids are the joining of two different species within an orchid genus. Paph. Maudiae is a primary hybrid (callosum x lawrenceanum). The backcross (lawrenceanum x Maudiae) is Alma Gevaert, which contains just the same two species, so we get the term "near primary" hybrid. This can go on forever as Paph. (Alma Gevaert x Maudiae) became Paph Almaud, still containing the same two species. Somewhere in here the term "Maudiae-type" hybrid probably got coined to convey a particular shape/coloration theme to a cross.
The entrance of a third species is probably what crosses the bridge into "complex land". Paph (curtisii x Maudiae) is Paph Emerald and (Emerald x Maudiae) became the famous hybrid Paph Claire de Lune. I think of this as a "Maudiae-type", but "complex" hybrid. But the term "Maudiae" isn't even specific to Paph, since there is a Cymbidium Maudiae and an Oncidopsis Maudiae!
The first time the name "Bulldog" was used with an orchid hybrid was 1979 in Paph British Bulldog, which was (Small World x Winston Churchill), but subsequently there was a Cymbidium Bulldog and an Oncidium Bulldog, so the term "Bulldog" isn't really specific to Paphs. But, Paph Winston Churchill was an incredibly famous and important Paph hybrid registered in 1951 and Sir Churchill's countenance and attitude was often called "bulldog like". Winston Churchill already contained 6 different species, so it was very complex, but the term "bulldog Paph" came to refer to large, mostly round, usually spotted, and usually red/brown colored complex Paph hybrids.
Keeping it simple, I think of complex hybrids of any orchid genus as containing 3 or more different species. I think of Maudiae-type and Bulldog-type Paphs as describing a particular shape and coloration of a Paph hybrid. There are growers who lean strongly toward only wanting to have species or primary hybrids in a particular genera, while others may care only about having the largest, most colorful hybrids, which may mean a preponderance of very complex hybrids, usually with some tetraploid things thrown in. Even by the 1940s, every large, white Phalaenopsis contained at least 3 species, so was complex.
I just bloomed Rlc. Yoshiko, made by Orchids Limited, and this Cattleya-alliance hybrid contains a mix of 21 different species, and I am sure that other hybrids can cram in even more species.
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