There are two senses of "ideology": The first sense (ideology) just denotes a set of ideas consciously and intentionally expressed, related and somehow bounded. The second sense (dogmatism) denotes a set of ideas adhered to dogmatically or inflexibly. You cannot have dogmatism without ideology, something specific and bounded to be dogmatic about, so these two senses get confused.
It seems like a clever panacea to eliminate dogmatism by eliminating ideology: it's true that you can't be dogmatic without something to be dogmatic about. But this approach throws out the baby with the bathwater.
It is possible to eliminate ideology, but to do so eliminates being intentional and conscious about your moral, ethical and political beliefs. Once you become conscious of your moral beliefs, you begin constructing an ideology. Once you begin sharing your moral beliefs and trying to persuade others to their value, you begin creating a socially constructed ideology.
There is a set of beliefs underlying all social constructs: they exist whether we discuss them consciously or just adhere to them unconsciously. And there are specific moral, ethical and political ideas underlying the status quo, how we live right now. To criticize these existing beliefs, we must become conscious of them, and accept some of them on their merits, and reject others and hold something different on their merits. To be critical of the status quo, we must begin constructing an ideology.
To profess that one is "non-ideological" then means accepting the status quo uncritically. Lack of criticality is an essential feature of dogmatism, so professing a lack of ideology shares the most objectionable feature of dogmatism.