Pronunciation of 这 (ze4 or zen4?) in 天气这么好 Not entirely sure, but a friend (mother tongue 普通话, originally from 沈阳(Shenyang)) recently pronounced 这 as ze4 or maybe zen4 when she said 今天天气这么好. 

这 isn't such simple word I find, as it can be zhe4 or zhei4, but I was not aware of any other ways it could be pronounced. Is ze4 or (zen4?) a common alternative?
'zhe4' is the most formal pronunciation. 'zhei4', 'zen4', and 'zhen4' are informal and dialectal.

I grew up in Beijing and local people use these variants frequently. (In contrast, 'zhe4' is actually not often heard in informal speech in the region.) In particular, 'zhei4' is common when people use 这 as a pronoun by itself (e.g., 这位 (this person)、这边儿 (this way)), and 'zen4' and 'zhen4' (less common than 'zen4') appear exclusively in 这么. Compare 怎么 'zen3me'.

The pronunciation of 'ze4' is not heard in the Beijing dialect, but could appear in other dialects of Mandarin.

Related note: When used alone to indicate location, 这儿, 'zher4' in pinyin, sounds more like 'zhar4' in the Beijing dialect - that is about somewhere between jerk sans k and jar in Midwestern American English.

Fun note: In very, very native Beijing dialect, 这儿 is sometimes pronounced like 'zhar4-har'. It might not be completely intelligible to people who grew up in regions in China farther away from Beijing.